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Difference between revisions of "Congruence"

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A relation between two integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248502.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248503.png" />, signifying that the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248504.png" /> between them is divisible by a given positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248505.png" />, which is called the modulus (or module) of the congruence; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248506.png" /> is then called a remainder of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248507.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248508.png" /> (cf. [[Remainder of an integer|Remainder of an integer]]). In order to express the congruence of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c0248509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485010.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485011.png" />, the symbol
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485012.png" /></td> </tr></table>
+
{{MSC|11A07}}
  
is used. If the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485013.png" /> is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485016.png" /> are said to be incongruent modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485017.png" />, and in order to express the incongruency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485019.png" />, the symbol
+
A relation between two integers  $  a $
 +
and  $  b $
 +
of the form  $  a = b + mk $,
 +
signifying that the difference $  a-b $
 +
between them is divisible by a given positive integer  $  m $,  
 +
which is called the modulus (or module) of the congruence;  $  a $
 +
is then called a remainder of  $  b $
 +
modulo $  m $(
 +
cf. [[Remainder of an integer|Remainder of an integer]]). In order to express the congruence of the numbers  $  a $
 +
and $  b $
 +
modulo  $  m $,  
 +
the symbol
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485020.png" /></td> </tr></table>
+
$$
 +
a \  \equiv \  b \  (  \mathop{\rm mod}\nolimits \  m)
 +
$$
  
is used. The congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485021.png" /> expresses that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485023.png" /> have identical remainders when divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485024.png" />.
+
is used. If the difference  $  a-b $
 +
is not divisible by  $  m $,
 +
then  $  a $
 +
and  $  b $
 +
are said to be incongruent modulo  $  m $,
 +
and in order to express the incongruency of  $  a $
 +
and $  b $,
 +
the symbol
  
Congruence modulo a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485025.png" /> is an equivalence relation: it is reflexive, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485027.png" />); symmetric, since it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485028.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485029.png" />) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485030.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485031.png" />); and transitive, since it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485032.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485033.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485035.png" />) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485036.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485037.png" />). Therefore, the relation " (modm)" divides the set of all integers into non-intersecting equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485038.png" />. Two integers belong to one and the same class if and only if they are congruent modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485039.png" />. These classes are called residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485041.png" />. Every integer is congruent modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485042.png" /> with just one of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485043.png" />; the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485044.png" /> belong to different classes, so that there are exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485045.png" /> residue classes, while the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485046.png" /> form a set of representatives of these classes. Any set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485047.png" /> integers containing one number from each residue class is called a complete residue system modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485049.png" />.
+
$$
 +
a \  \not\equiv \  b \  (  \mathop{\rm mod}\nolimits \  m)
 +
$$
 +
 
 +
is used. The congruence  $  a \equiv b \  (  \mathop{\rm mod}\nolimits \  m) $
 +
expresses that  $  a $
 +
and  $  b $
 +
have identical remainders when divided by  $  m $.
 +
 
 +
Congruence modulo a fixed $  m $
 +
is an [[equivalence relation]]: it is reflexive, since $  a \equiv a $(
 +
$  \mathop{\rm mod}\nolimits \  m $);  
 +
symmetric, since it follows from $  a \equiv b $(
 +
$  \mathop{\rm mod}\nolimits \  m $)  
 +
that $  b \equiv a $(
 +
$  \mathop{\rm mod}\nolimits \  m $);  
 +
and transitive, since it follows from $  a \equiv b $(
 +
$  \mathop{\rm mod}\nolimits \  m $)  
 +
and $  b \equiv c $(
 +
$  \mathop{\rm mod}\nolimits \  m $)  
 +
that $  a \equiv c $(
 +
$  \mathop{\rm mod}\nolimits \  m $).  
 +
Therefore, the relation " (modm)" divides the set of all integers into non-intersecting equivalence classes $  A,\  B ,\dots $.  
 +
Two integers belong to one and the same class if and only if they are congruent modulo $  m $.  
 +
These classes are called residue classes modulo $  m $.  
 +
Every integer is congruent modulo $  m $
 +
with just one of the numbers 0 \dots m-1 $;  
 +
the numbers 0 \dots m-1 $
 +
belong to different classes, so that there are exactly $  m $
 +
residue classes, while the numbers 0 \dots m-1 $
 +
form a set of representatives of these classes. Any set of $  m $
 +
integers containing one number from each residue class is called a complete residue system modulo $  m $.
  
 
Congruences modulo one and the same number can be added, subtracted and multiplied in the same way as normal equalities, i.e. it follows from
 
Congruences modulo one and the same number can be added, subtracted and multiplied in the same way as normal equalities, i.e. it follows from
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485050.png" /></td> </tr></table>
+
$$
 +
a \  \equiv \  b \  (  \mathop{\rm mod}\nolimits \  m) \ \ 
 +
\textrm{ and } \ \  c \  \equiv \  d \  (  \mathop{\rm mod}\nolimits \  m)
 +
$$
  
 
that
 
that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485051.png" /></td> </tr></table>
+
$$
 +
a \pm c \  \equiv \  b \pm d \  (  \mathop{\rm mod}\nolimits \  m) \ \ 
 +
\textrm{ and } \ \
 +
ac \  \equiv \  bd \  (  \mathop{\rm mod}\nolimits \  m).
 +
$$
  
Both parts of a congruence can be multiplied by one and the same integer, and both can be divided by a common divisor, as long as the divisor is relatively prime with the modulus of the congruence. If a common divisor of a number by which both parts of the congruence are divisible and the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485052.png" /> of the congruence is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485053.png" />, then after dividing by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485054.png" /> a congruence modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485055.png" /> is obtained.
+
Both parts of a congruence can be multiplied by one and the same integer, and both can be divided by a common divisor, as long as the divisor is relatively prime with the modulus of the congruence. If a common divisor of a number by which both parts of the congruence are divisible and the modulus $  m $
 +
of the congruence is $  d $,  
 +
then after dividing by $  d $
 +
a congruence modulo $  m/d $
 +
is obtained.
  
The operations of addition, subtraction and multiplication of congruences induce similar operations on the residue classes. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485057.png" /> are arbitrary elements from the residue classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485059.png" />, respectively, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485060.png" /> always belongs to one and the same residue class, called the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485061.png" /> of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485063.png" />. The difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485064.png" /> and the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485065.png" /> of the two residue classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485067.png" /> are defined in the same way. The residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485068.png" /> form an Abelian group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485069.png" /> with respect to addition. The zero element of this group is the class consisting of all integers that are multiples of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485070.png" />; the negative (inverse) of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485071.png" /> is the class consisting of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485072.png" /> with a minus sign. Moreover, the residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485073.png" /> form a ring relative to the above-defined operations of addition, subtraction and multiplication.
+
The operations of addition, subtraction and multiplication of congruences induce similar operations on the residue classes. Thus, if $  a $
 +
and $  b $
 +
are arbitrary elements from the residue classes $  A $
 +
and $  B $,  
 +
respectively, then $  a+b $
 +
always belongs to one and the same residue class, called the sum $  A+B $
 +
of the classes $  A $
 +
and $  B $.  
 +
The difference $  A-B $
 +
and the product $  A \cdot B $
 +
of the two residue classes $  A $
 +
and $  B $
 +
are defined in the same way. The residue classes modulo $  m $
 +
form an Abelian group of order $  m $
 +
with respect to addition. The zero element of this group is the class consisting of all integers that are multiples of the number $  m $;  
 +
the negative (inverse) of a class $  A $
 +
is the class consisting of all elements of $  A $
 +
with a minus sign. Moreover, the residue classes modulo $  m $
 +
form a ring relative to the above-defined operations of addition, subtraction and multiplication.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485074.png" /> be a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485075.png" /> variables with integer coefficients. An equation of the form
+
Let $  F(x _{1} \dots x _{n} ) $
 +
be a polynomial in $  n $
 +
variables with integer coefficients. An equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{*}
 +
F(x _{1} \dots x _{n} ) \  \equiv \  0 ( \mathop{\rm mod}\nolimits \  m)
 +
$$
  
is called a [[Congruence equation|congruence equation]]. A solution of the congruence (*) is any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485077.png" /> of integer values of the unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485078.png" /> such that the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485079.png" /> is congruent to zero modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485080.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485082.png" />, belong to the residue classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485083.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485084.png" />, then any other set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485086.png" />, is also a solution of the congruence. One therefore calls the set of residue classes itself, of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485087.png" /> are representatives, a solution of the congruence (*), i.e. a solution of the algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485088.png" /> in the ring of residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485089.png" />. Given this manner of speaking, the congruence (*) has as many solutions as there are sets of residue classes of a complete system modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485090.png" /> that satisfy the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485091.png" />. The number of solutions of more general forms of congruences, as well as of systems of congruences, is defined in the same way.
+
is called a [[Congruence equation|congruence equation]]. A solution of the congruence (*) is any set $  a _{1} \dots a _{n} $
 +
of integer values of the unknowns $  x _{1} \dots x _{n} $
 +
such that the number $  F(a _{1} \dots a _{n} ) $
 +
is congruent to zero modulo $  m $.  
 +
If $  a _{i} $,  
 +
$  1 \leq  i \leq  n $,  
 +
belong to the residue classes $  X _{i} $
 +
modulo $  m $,  
 +
then any other set $  a _ i^ \prime  \in X _{i} $,  
 +
$  1 \leq  i \leq  n $,  
 +
is also a solution of the congruence. One therefore calls the set of residue classes itself, of which $  a _{1} \dots a _{n} $
 +
are representatives, a solution of the congruence (*), i.e. a solution of the algebraic equation $  F(x _{1} \dots x _{n} ) = 0 $
 +
in the ring of residue classes modulo $  m $.  
 +
Given this manner of speaking, the congruence (*) has as many solutions as there are sets of residue classes of a complete system modulo $  m $
 +
that satisfy the equation $  F(x _{1} \dots x _{n} ) = 0 $.  
 +
The number of solutions of more general forms of congruences, as well as of systems of congruences, is defined in the same way.
  
 
A congruence of the first degree
 
A congruence of the first degree
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485092.png" /></td> </tr></table>
+
$$
 +
ax \  \equiv \  b \  (  \mathop{\rm mod}\nolimits \  m),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485093.png" /> is relatively prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485094.png" /> (this is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485095.png" />), has a unique solution. Those residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485096.png" /> whose elements are relatively prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485097.png" /> form an Abelian group with respect to multiplication. The unit of this group is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485098.png" /> which contains the number 1, while the inverse of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c02485099.png" /> is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850100.png" /> that contains the solutions of the congruence
+
where $  a $
 +
is relatively prime with $  m $(
 +
this is denoted by $  (a,\  m) = 1 $),  
 +
has a unique solution. Those residue classes modulo $  m $
 +
whose elements are relatively prime with $  m $
 +
form an Abelian group with respect to multiplication. The unit of this group is the class $  E $
 +
which contains the number 1, while the inverse of a class $  A $
 +
is the class $  A^{-1} $
 +
that contains the solutions of the congruence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850101.png" /></td> </tr></table>
+
$$
 +
ax \  \equiv \  1 \  (  \mathop{\rm mod}\nolimits \  m),\ \  \textrm{ where } \  a \in A.
 +
$$
  
The residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850102.png" /> consisting of elements which are relatively prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850103.png" /> are called reduced classes, while any system of numbers containing one number from each reduced class is called a reduced residue system. A reduced residue system consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850104.png" /> elements, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850105.png" /> is the [[Euler function|Euler function]], which is equal to the number of elements in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850106.png" /> that are relatively prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850107.png" />.
+
The residue classes modulo $  m $
 +
consisting of elements which are relatively prime with $  m $
 +
are called reduced classes, while any system of numbers containing one number from each reduced class is called a reduced residue system. A reduced residue system consists of $  \phi (m) $
 +
elements, where $  \phi (m) $
 +
is the [[Euler function|Euler function]], which is equal to the number of elements in the set $  1 \dots m $
 +
that are relatively prime with $  m $.
  
The structure of the multiplicative group of reduced residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850108.png" /> is explained to a significant degree by the theorems of L. Euler and P. Fermat, which show that the order of any element of the group divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850109.png" />.
+
The structure of the multiplicative group of reduced residue classes modulo $  m $
 +
is explained to a significant degree by the theorems of L. Euler and P. Fermat, which show that the order of any element of the group divides $  \phi (m) $.
  
Euler's theorem. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850111.png" /> are relatively prime, then
+
Euler's theorem. If $  a $
 +
and $  m $
 +
are relatively prime, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850112.png" /></td> </tr></table>
+
$$
 +
a ^ {\phi (m)} \  \equiv \  1 \  (  \mathop{\rm mod}\nolimits \  m)
 +
$$
  
 
(see [[Euler theorem|Euler theorem]]).
 
(see [[Euler theorem|Euler theorem]]).
  
The particular case of this theorem for a prime modulus is Fermat's theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850113.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850114.png" /> is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850115.png" />, then
+
The particular case of this theorem for a prime modulus is Fermat's theorem: If $  p $
 +
is a prime number and $  a $
 +
is not divisible by $  p $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850116.png" /></td> </tr></table>
+
$$
 +
a^{p-1} \  \equiv \  1 \  (  \mathop{\rm mod}\nolimits \  p)
 +
$$
  
 
(see [[Fermat little theorem|Fermat little theorem]]).
 
(see [[Fermat little theorem|Fermat little theorem]]).
  
An important concept in the study of the multiplicative group of reduced residue classes is the concept of a [[Primitive root|primitive root]] modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850117.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850118.png" />, there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850119.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850120.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850121.png" />), for example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850122.png" />. The least of these is called the index to which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850123.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850124.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850125.png" /> belongs.
+
An important concept in the study of the multiplicative group of reduced residue classes is the concept of a [[Primitive root|primitive root]] modulo $  m $.  
 +
When $  (a,\  m) = 1 $,  
 +
there exists a positive integer $  \gamma $
 +
for which $  a^ \gamma  \equiv 1 $(
 +
$  \mathop{\rm mod}\nolimits \  m $),  
 +
for example $  \gamma = \phi (m) $.  
 +
The least of these is called the index to which the $  number $
 +
$  a $
 +
modulo $  m $
 +
belongs.
  
Numbers which belong to the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850126.png" /> (if there are any) are called primitive roots modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850128.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850129.png" /> is a primitive root modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850131.png" /> runs through a complete residue system modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850132.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850133.png" /> runs through a reduced residue system modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850134.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850135.png" />, then for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850136.png" /> from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850137.png" />, the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850138.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850139.png" />) holds. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850140.png" /> is called the index of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850143.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850144.png" /> with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850145.png" /> and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850146.png" /> or, more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850147.png" />. The properties of an index are broadly the same as the properties of a logarithm. Primitive roots exist only for moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850148.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850149.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850150.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850151.png" /> is an integer. In these cases, the multiplicative group of reduced residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850152.png" /> is simply the cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850153.png" />. In the other cases, the groups of reduced residue classes have a much more complex structure.
+
Numbers which belong to the index $  \phi (m) $(
 +
if there are any) are called primitive roots modulo $  m $.  
 +
If $  g $
 +
is a primitive root modulo $  m $
 +
and $  \gamma $
 +
runs through a complete residue system modulo $  \phi (m) $,  
 +
then $  g^ \gamma  $
 +
runs through a reduced residue system modulo $  m $.  
 +
Thus, if $  (a,\  m) = 1 $,  
 +
then for a certain $  \gamma $
 +
from the set 0 \dots \phi (m)-1 $,  
 +
the congruence $  a \equiv g^ \gamma  $(
 +
$  \mathop{\rm mod}\nolimits \  m $)  
 +
holds. This $  \gamma $
 +
is called the index of the number $  a $
 +
modulo $  m $
 +
with respect to the basis $  g $
 +
and is denoted by the symbol $  \mathop{\rm ind}\nolimits \  a $
 +
or, more precisely, $  \mathop{\rm ind}\nolimits _{g} \  a $.  
 +
The properties of an index are broadly the same as the properties of a logarithm. Primitive roots exist only for moduli $  m > 1 $
 +
of the form $  2,\  4,\  p^ \alpha  ,\  2p^ \alpha  $,  
 +
where $  p \geq 3 $
 +
is a prime number and $  \alpha \geq 1 $
 +
is an integer. In these cases, the multiplicative group of reduced residue classes modulo $  m $
 +
is simply the cyclic group of order $  \phi (m) $.  
 +
In the other cases, the groups of reduced residue classes have a much more complex structure.
  
Many problems in number theory reduce to the question of the solvability or unsolvability of some type of congruence. As a result of this, the theory of congruences, which was first systematically developed by C.F. Gauss (see [[#References|[5]]]) and used by him as a foundation of classical number theory, is to this day one of the basic means of solving number-theoretical problems. In this connection, research into the question of the number of solutions of a congruence equation is of fundamental importance to number theory. The simplest types of congruence equations are congruences of the first degree with one unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850154.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850155.png" />). A complete answer to the question of the number of solutions of congruences of the first degree is provided by the following theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850156.png" />. The congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850157.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850158.png" />) is unsolvable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850159.png" /> is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850160.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850161.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850162.png" />, the congruence has precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850163.png" /> solutions.
+
Many problems in number theory reduce to the question of the solvability or unsolvability of some type of congruence. As a result of this, the theory of congruences, which was first systematically developed by C.F. Gauss (see [[#References|[5]]]) and used by him as a foundation of classical number theory, is to this day one of the basic means of solving number-theoretical problems. In this connection, research into the question of the number of solutions of a congruence equation is of fundamental importance to number theory. The simplest types of congruence equations are congruences of the first degree with one unknown $  ax \equiv b $(
 +
$  \mathop{\rm mod}\nolimits \  m $).  
 +
A complete answer to the question of the number of solutions of congruences of the first degree is provided by the following theorem. Let $  (a,\  m) = d $.  
 +
The congruence $  ax \equiv b $(
 +
$  \mathop{\rm mod}\nolimits \  m $)  
 +
is unsolvable if $  b $
 +
is not divisible by $  d $.  
 +
When $  b $
 +
is a multiple of $  d $,  
 +
the congruence has precisely $  d $
 +
solutions.
  
 
The question of the solvability of a system of linear congruences
 
The question of the solvability of a system of linear congruences
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850164.png" /></td> </tr></table>
+
$$
 +
\sum _ { j=1 } ^ s a _{ij} x _{j} \  \equiv \  b _{i} \  (  \mathop{\rm mod}\nolimits \  p),\ \
 +
i = 1 \dots t,
 +
$$
  
modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850165.png" /> is solved completely in the general theory of linear equations over arbitrary fields. The case of a composite modulus can be reduced to the case of a prime modulus.
+
modulo a prime number $  p > 2 $
 +
is solved completely in the general theory of linear equations over arbitrary fields. The case of a composite modulus can be reduced to the case of a prime modulus.
  
 
The form of algebraic congruences with one unknown which is the next most complex to study is a [[Two-term congruence|two-term congruence]]
 
The form of algebraic congruences with one unknown which is the next most complex to study is a [[Two-term congruence|two-term congruence]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850166.png" /></td> </tr></table>
+
$$
 +
x^{n} \  \equiv \  a \  (  \mathop{\rm mod}\nolimits \  m) \ \  \textrm{ when } \  (a,\  m) = 1.
 +
$$
  
If the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850167.png" /> has solutions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850168.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850171.png" />-th power residue modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850172.png" />; if it does not, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850173.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850176.png" />-th power non-residue modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850177.png" />. More specifically, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850178.png" />, the residues or non-residues are called quadratic, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850179.png" />, cubic, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850180.png" />, bi-quadratic.
+
If the congruence $  x^{n} \equiv a \  (  \mathop{\rm mod}\nolimits \  m) $
 +
has solutions, then $  a $
 +
is called an $  n $-
 +
th power residue modulo $  m $;  
 +
if it does not, then $  a $
 +
is called an $  n $-
 +
th power non-residue modulo $  m $.  
 +
More specifically, when $  n=2 $,  
 +
the residues or non-residues are called quadratic, when $  n=3 $,  
 +
cubic, and when $  n=4 $,  
 +
bi-quadratic.
  
 
The question of the number of solutions of a congruence
 
The question of the number of solutions of a congruence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850181.png" /></td> </tr></table>
+
$$
 +
f(x) \  \equiv \  0 \  (  \mathop{\rm mod}\nolimits \  m)
 +
$$
  
modulo a composite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850182.png" /> reduces to the question of the number of solutions of the congruence
+
modulo a composite number $  m = p _{1} ^ {\alpha _ 1} \dots p _{s} ^ {\alpha _ s} $
 +
reduces to the question of the number of solutions of the congruence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850183.png" /></td> </tr></table>
+
$$
 +
f(x) \  \equiv \  0 \  (  \mathop{\rm mod}\nolimits \  p _{i} ^ {\alpha _ i} )
 +
$$
  
for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850184.png" />. The following theorem holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850185.png" /> are pairwise relatively prime (coprime) positive integers, then the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850186.png" /> of solutions of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850187.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850188.png" />) can be expressed by the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850189.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850190.png" />, which are equal to the number of solutions of the corresponding congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850191.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850192.png" />) according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850193.png" />. In its turn, the question of the number of solutions of the congruence
+
for every $  i = 1 \dots s $.  
 +
The following theorem holds: If $  m _{1} \dots m _{r} $
 +
are pairwise relatively prime (coprime) positive integers, then the number $  N $
 +
of solutions of the congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  m _{1} \dots m _{r} $)  
 +
can be expressed by the values $  N _{i} $,  
 +
$  i = 1 \dots r $,  
 +
which are equal to the number of solutions of the corresponding congruences $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  m _{i} $)  
 +
according to the formula $  N = \prod _ i=1^{r} N _{i} $.  
 +
In its turn, the question of the number of solutions of the congruence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850194.png" /></td> </tr></table>
+
$$
 +
f(x) \  \equiv \  0 \  (  \mathop{\rm mod}\nolimits \  p^ \alpha  ),\ \
 +
\alpha > 1 ,
 +
$$
  
generally reduces to the question of the number of solutions of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850195.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850196.png" />) where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850197.png" /> is a prime number. The following theorem forms the basis of this reduction: Every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850198.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850199.png" />) of a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850200.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850201.png" />) can be uniquely continued to a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850202.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850203.png" />) of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850204.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850205.png" />) provided that the formal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850206.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850207.png" />), i.e. a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850208.png" /> of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850209.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850210.png" />) then exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850211.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850212.png" />).
+
generally reduces to the question of the number of solutions of the congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
where $  p $
 +
is a prime number. The following theorem forms the basis of this reduction: Every solution $  x \equiv x _{1} $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
of a congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
can be uniquely continued to a solution $  x \equiv x _ \alpha  $(
 +
$  \mathop{\rm mod}\nolimits \  p^ \alpha  $)  
 +
of the congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p^ \alpha  $)  
 +
provided that the formal derivative $  f ^ {\  \prime} (x _{1} ) \not\equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $),  
 +
i.e. a unique solution $  x _ \alpha  $
 +
of the congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p^ \alpha  $)  
 +
then exists such that $  x _ \alpha  \equiv x _{1} $(
 +
$  \mathop{\rm mod}\nolimits \  p $).
  
A similar situation also arises in the case of a congruence equation in several variables, i.e. in non-degenerate cases the question of the number of solutions of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850213.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850214.png" />), modulo a composite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850215.png" /> reduces to the question of the number of solutions of the same congruence modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850216.png" />.
+
A similar situation also arises in the case of a congruence equation in several variables, i.e. in non-degenerate cases the question of the number of solutions of the congruence $  F(x _{1} \dots x _{n} ) = 0 $(
 +
$  \mathop{\rm mod}\nolimits \  m $),  
 +
modulo a composite number $  m $
 +
reduces to the question of the number of solutions of the same congruence modulo a prime number $  m $.
  
An upper bound for the number of solutions of a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850217.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850218.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850219.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850220.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850221.png" />), is given by Lagrange's theorem: The number of solutions of a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850222.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850223.png" />) does not exceed the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850224.png" />. The study of the question of the exact number of solutions of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850225.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850226.png" />) generally meets with considerable difficulties, so that only a few results have so far (1984) been obtained in this area.
+
An upper bound for the number of solutions of a congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $),  
 +
where $  f(x) = a _{0} x^{n} + \dots + a _{n} $,  
 +
$  a _{0} \not\equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $),  
 +
is given by Lagrange's theorem: The number of solutions of a congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
does not exceed the degree of the polynomial $  f(x) $.  
 +
The study of the question of the exact number of solutions of the congruence $  f(x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
generally meets with considerable difficulties, so that only a few results have so far (1984) been obtained in this area.
  
The Lagrange theorem ceases to hold in the case of a composite modulus. The above-mentioned specific property of prime numbers is explained by the fact that the residue classes modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850227.png" /> form a field (see [[Congruence modulo a prime number|Congruence modulo a prime number]]). Another specific property of prime numbers which is explained by this fact is expressed in the [[Wilson theorem|Wilson theorem]].
+
The Lagrange theorem ceases to hold in the case of a composite modulus. The above-mentioned specific property of prime numbers is explained by the fact that the residue classes modulo a prime number $  p $
 +
form a field (see [[Congruence modulo a prime number|Congruence modulo a prime number]]). Another specific property of prime numbers which is explained by this fact is expressed in the [[Wilson theorem|Wilson theorem]].
  
 
In the study of quadratic congruences
 
In the study of quadratic congruences
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850228.png" /></td> </tr></table>
+
$$
 +
x^{2} \  \equiv \  a \  (  \mathop{\rm mod}\nolimits \  p)
 +
$$
  
modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850229.png" />, as well as in their applications, the [[Legendre symbol|Legendre symbol]] and the [[Jacobi symbol|Jacobi symbol]] are introduced.
+
modulo a prime number $  p $,  
 +
as well as in their applications, the [[Legendre symbol|Legendre symbol]] and the [[Jacobi symbol|Jacobi symbol]] are introduced.
  
 
Congruence equations modulo a prime number in two unknowns (and generally in any number of unknowns),
 
Congruence equations modulo a prime number in two unknowns (and generally in any number of unknowns),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850230.png" /></td> </tr></table>
+
$$
 +
F(x,\  y) \  \equiv \  0 \  (  \mathop{\rm mod}\nolimits \  p) ,
 +
$$
  
can be treated as equations over the finite prime field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850231.png" /> elements. Methods of the theory of algebraic functions and algebraic geometry are therefore used in studying them, as well as methods of number theory. Using these methods, an asymptotic expansion in the form
+
can be treated as equations over the finite prime field of $  p $
 +
elements. Methods of the theory of algebraic functions and algebraic geometry are therefore used in studying them, as well as methods of number theory. Using these methods, an asymptotic expansion in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850232.png" /></td> </tr></table>
+
$$
 +
N _{p} \  = \  p + O( \sqrt p )
 +
$$
  
was first obtained for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850233.png" /> of solutions of a broad class of congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850234.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850235.png" />). The same asymptotic expansions were subsequently deduced using methods of number theory without going beyond the limits of the theory of congruences (see [[#References|[6]]]).
+
was first obtained for the number $  N _{p} $
 +
of solutions of a broad class of congruences $  F(x,\  y) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $).  
 +
The same asymptotic expansions were subsequently deduced using methods of number theory without going beyond the limits of the theory of congruences (see [[#References|[6]]]).
  
Comparatively little is known about algebraic congruences in several unknowns. The following can be considered as a general result (Chevalley's theorem). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850236.png" /> be a polynomial with integer rational coefficients, the degree of which is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850237.png" />. The number of solutions of the congruence
+
Comparatively little is known about algebraic congruences in several unknowns. The following can be considered as a general result (Chevalley's theorem). Let $  F(x _{1} \dots x _{n} ) $
 +
be a polynomial with integer rational coefficients, the degree of which is less than $  n $.  
 +
The number of solutions of the congruence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850238.png" /></td> </tr></table>
+
$$
 +
F(x _{1} \dots x _{n} ) \  \equiv \  0 \  (  \mathop{\rm mod}\nolimits \  p)
 +
$$
  
is then divisible by the prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850239.png" />. One very strong result for this type of questions was obtained by P. Deligne [[#References|[9]]] (see also [[#References|[10]]]).
+
is then divisible by the prime number $  p $.  
 +
One very strong result for this type of questions was obtained by P. Deligne [[#References|[9]]] (see also [[#References|[10]]]).
  
Great value is also attached to research into the question of the number of solutions of congruences in an incomplete residue system. A systematic solution to this type of problems was first attempted by I.M. Vinogradov (see [[#References|[4]]]). An example of this type of problems is the problem of the distribution of quadratic residues and non-residues in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850240.png" />, which is connected with the study of solutions of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850241.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850242.png" />) (see [[Vinogradov hypotheses|Vinogradov hypotheses]]; [[Distribution of power residues and non-residues|Distribution of power residues and non-residues]]).
+
Great value is also attached to research into the question of the number of solutions of congruences in an incomplete residue system. A systematic solution to this type of problems was first attempted by I.M. Vinogradov (see [[#References|[4]]]). An example of this type of problems is the problem of the distribution of quadratic residues and non-residues in the set $  1 \dots p-1 $,  
 +
which is connected with the study of solutions of the congruence $  y^{2} \equiv x $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
(see [[Vinogradov hypotheses|Vinogradov hypotheses]]; [[Distribution of power residues and non-residues|Distribution of power residues and non-residues]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0265267}} {{ZBL|0204.37101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.M. Vinogradov, "Fundamentals of number theory" , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) {{MR|0807530}} {{ZBL|0577.01049}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin) {{MR|0188045}} {{ZBL|21.0166.04}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.A. Stepanov, "A constructive method in the theory of equations over finite fields" ''Trudy Mat. Inst. Steklov.'' , '''132''' (1973) pp. 237–246 (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) {{MR|0153659}} {{ZBL|1038.11500}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) {{MR|0249348}} {{ZBL|0169.37502}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 {{MR|0340258}} {{ZBL|0314.14007}} {{ZBL|0287.14001}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> N.M. Katz, "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields" F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert problems'' , ''Proc. Symp. Pure Math.'' , '''28''' , Amer. Math. Soc. (1976) pp. 275–305</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0265267}} {{ZBL|0204.37101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.M. Vinogradov, "Fundamentals of number theory" , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) {{MR|0807530}} {{ZBL|0577.01049}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin) {{MR|0188045}} {{ZBL|21.0166.04}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.A. Stepanov, "A constructive method in the theory of equations over finite fields" ''Trudy Mat. Inst. Steklov.'' , '''132''' (1973) pp. 237–246 (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) {{MR|0153659}} {{ZBL|1038.11500}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) {{MR|0249348}} {{ZBL|0169.37502}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 {{MR|0340258}} {{ZBL|0314.14007}} {{ZBL|0287.14001}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> N.M. Katz, "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields" F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert problems'' , ''Proc. Symp. Pure Math.'' , '''28''' , Amer. Math. Soc. (1976) pp. 275–305</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The simultaneous system of congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850243.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850244.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850245.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850246.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850247.png" />, has a unique residue class modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850248.png" /> as solution. This is the content of the [[Chinese remainder theorem|Chinese remainder theorem]].
+
The simultaneous system of congruences $  x \equiv c _{i} $(
 +
$  \mathop{\rm mod}\nolimits \  m _{i} $),  
 +
$  i = 1 \dots r $,  
 +
where $  ( m _{i} ,\  m _{j} ) = 1 $
 +
for all $  i \neq j $,  
 +
has a unique residue class modulo $  M = m _{1} \dots m _{r} $
 +
as solution. This is the content of the [[Chinese remainder theorem|Chinese remainder theorem]].
  
The procedure to extend solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850249.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850250.png" />) to solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850251.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850252.png" />) is described in the [[Hensel lemma|Hensel lemma]] (cf. [[#References|[a2]]]), and extends to a technique in the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850253.png" />-adic numbers (cf. [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850254.png" />-adic number]]).
+
The procedure to extend solutions of $  f (x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p $)  
 +
to solutions of $  f (x) \equiv 0 $(
 +
$  \mathop{\rm mod}\nolimits \  p^ \alpha  $)  
 +
is described in the [[Hensel lemma|Hensel lemma]] (cf. [[#References|[a2]]]), and extends to a technique in the theory of $  p $-
 +
adic numbers (cf. [[P-adic number| $  p $-
 +
adic number]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Koblitz, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850255.png" />-adic numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850256.png" />-adic analysis and zeta-functions" , Springer (1977) pp. Chapt. 1 {{MR|466081}} {{ZBL|}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Koblitz, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850255.png" />-adic numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024850/c024850256.png" />-adic analysis and zeta-functions" , Springer (1977) pp. Chapt. 1 {{MR|466081}} {{ZBL|}} </TD></TR>
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</table>

Latest revision as of 09:30, 1 February 2020


2010 Mathematics Subject Classification: Primary: 11A07 [MSN][ZBL]

A relation between two integers $ a $ and $ b $ of the form $ a = b + mk $, signifying that the difference $ a-b $ between them is divisible by a given positive integer $ m $, which is called the modulus (or module) of the congruence; $ a $ is then called a remainder of $ b $ modulo $ m $( cf. Remainder of an integer). In order to express the congruence of the numbers $ a $ and $ b $ modulo $ m $, the symbol

$$ a \ \equiv \ b \ ( \mathop{\rm mod}\nolimits \ m) $$

is used. If the difference $ a-b $ is not divisible by $ m $, then $ a $ and $ b $ are said to be incongruent modulo $ m $, and in order to express the incongruency of $ a $ and $ b $, the symbol

$$ a \ \not\equiv \ b \ ( \mathop{\rm mod}\nolimits \ m) $$

is used. The congruence $ a \equiv b \ ( \mathop{\rm mod}\nolimits \ m) $ expresses that $ a $ and $ b $ have identical remainders when divided by $ m $.

Congruence modulo a fixed $ m $ is an equivalence relation: it is reflexive, since $ a \equiv a $( $ \mathop{\rm mod}\nolimits \ m $); symmetric, since it follows from $ a \equiv b $( $ \mathop{\rm mod}\nolimits \ m $) that $ b \equiv a $( $ \mathop{\rm mod}\nolimits \ m $); and transitive, since it follows from $ a \equiv b $( $ \mathop{\rm mod}\nolimits \ m $) and $ b \equiv c $( $ \mathop{\rm mod}\nolimits \ m $) that $ a \equiv c $( $ \mathop{\rm mod}\nolimits \ m $). Therefore, the relation " (modm)" divides the set of all integers into non-intersecting equivalence classes $ A,\ B ,\dots $. Two integers belong to one and the same class if and only if they are congruent modulo $ m $. These classes are called residue classes modulo $ m $. Every integer is congruent modulo $ m $ with just one of the numbers $ 0 \dots m-1 $; the numbers $ 0 \dots m-1 $ belong to different classes, so that there are exactly $ m $ residue classes, while the numbers $ 0 \dots m-1 $ form a set of representatives of these classes. Any set of $ m $ integers containing one number from each residue class is called a complete residue system modulo $ m $.

Congruences modulo one and the same number can be added, subtracted and multiplied in the same way as normal equalities, i.e. it follows from

$$ a \ \equiv \ b \ ( \mathop{\rm mod}\nolimits \ m) \ \ \textrm{ and } \ \ c \ \equiv \ d \ ( \mathop{\rm mod}\nolimits \ m) $$

that

$$ a \pm c \ \equiv \ b \pm d \ ( \mathop{\rm mod}\nolimits \ m) \ \ \textrm{ and } \ \ ac \ \equiv \ bd \ ( \mathop{\rm mod}\nolimits \ m). $$

Both parts of a congruence can be multiplied by one and the same integer, and both can be divided by a common divisor, as long as the divisor is relatively prime with the modulus of the congruence. If a common divisor of a number by which both parts of the congruence are divisible and the modulus $ m $ of the congruence is $ d $, then after dividing by $ d $ a congruence modulo $ m/d $ is obtained.

The operations of addition, subtraction and multiplication of congruences induce similar operations on the residue classes. Thus, if $ a $ and $ b $ are arbitrary elements from the residue classes $ A $ and $ B $, respectively, then $ a+b $ always belongs to one and the same residue class, called the sum $ A+B $ of the classes $ A $ and $ B $. The difference $ A-B $ and the product $ A \cdot B $ of the two residue classes $ A $ and $ B $ are defined in the same way. The residue classes modulo $ m $ form an Abelian group of order $ m $ with respect to addition. The zero element of this group is the class consisting of all integers that are multiples of the number $ m $; the negative (inverse) of a class $ A $ is the class consisting of all elements of $ A $ with a minus sign. Moreover, the residue classes modulo $ m $ form a ring relative to the above-defined operations of addition, subtraction and multiplication.

Let $ F(x _{1} \dots x _{n} ) $ be a polynomial in $ n $ variables with integer coefficients. An equation of the form

$$ \tag{*} F(x _{1} \dots x _{n} ) \ \equiv \ 0 \ ( \mathop{\rm mod}\nolimits \ m) $$

is called a congruence equation. A solution of the congruence (*) is any set $ a _{1} \dots a _{n} $ of integer values of the unknowns $ x _{1} \dots x _{n} $ such that the number $ F(a _{1} \dots a _{n} ) $ is congruent to zero modulo $ m $. If $ a _{i} $, $ 1 \leq i \leq n $, belong to the residue classes $ X _{i} $ modulo $ m $, then any other set $ a _ i^ \prime \in X _{i} $, $ 1 \leq i \leq n $, is also a solution of the congruence. One therefore calls the set of residue classes itself, of which $ a _{1} \dots a _{n} $ are representatives, a solution of the congruence (*), i.e. a solution of the algebraic equation $ F(x _{1} \dots x _{n} ) = 0 $ in the ring of residue classes modulo $ m $. Given this manner of speaking, the congruence (*) has as many solutions as there are sets of residue classes of a complete system modulo $ m $ that satisfy the equation $ F(x _{1} \dots x _{n} ) = 0 $. The number of solutions of more general forms of congruences, as well as of systems of congruences, is defined in the same way.

A congruence of the first degree

$$ ax \ \equiv \ b \ ( \mathop{\rm mod}\nolimits \ m), $$

where $ a $ is relatively prime with $ m $( this is denoted by $ (a,\ m) = 1 $), has a unique solution. Those residue classes modulo $ m $ whose elements are relatively prime with $ m $ form an Abelian group with respect to multiplication. The unit of this group is the class $ E $ which contains the number 1, while the inverse of a class $ A $ is the class $ A^{-1} $ that contains the solutions of the congruence

$$ ax \ \equiv \ 1 \ ( \mathop{\rm mod}\nolimits \ m),\ \ \textrm{ where } \ a \in A. $$

The residue classes modulo $ m $ consisting of elements which are relatively prime with $ m $ are called reduced classes, while any system of numbers containing one number from each reduced class is called a reduced residue system. A reduced residue system consists of $ \phi (m) $ elements, where $ \phi (m) $ is the Euler function, which is equal to the number of elements in the set $ 1 \dots m $ that are relatively prime with $ m $.

The structure of the multiplicative group of reduced residue classes modulo $ m $ is explained to a significant degree by the theorems of L. Euler and P. Fermat, which show that the order of any element of the group divides $ \phi (m) $.

Euler's theorem. If $ a $ and $ m $ are relatively prime, then

$$ a ^ {\phi (m)} \ \equiv \ 1 \ ( \mathop{\rm mod}\nolimits \ m) $$

(see Euler theorem).

The particular case of this theorem for a prime modulus is Fermat's theorem: If $ p $ is a prime number and $ a $ is not divisible by $ p $, then

$$ a^{p-1} \ \equiv \ 1 \ ( \mathop{\rm mod}\nolimits \ p) $$

(see Fermat little theorem).

An important concept in the study of the multiplicative group of reduced residue classes is the concept of a primitive root modulo $ m $. When $ (a,\ m) = 1 $, there exists a positive integer $ \gamma $ for which $ a^ \gamma \equiv 1 $( $ \mathop{\rm mod}\nolimits \ m $), for example $ \gamma = \phi (m) $. The least of these is called the index to which the $ number $ $ a $ modulo $ m $ belongs.

Numbers which belong to the index $ \phi (m) $( if there are any) are called primitive roots modulo $ m $. If $ g $ is a primitive root modulo $ m $ and $ \gamma $ runs through a complete residue system modulo $ \phi (m) $, then $ g^ \gamma $ runs through a reduced residue system modulo $ m $. Thus, if $ (a,\ m) = 1 $, then for a certain $ \gamma $ from the set $ 0 \dots \phi (m)-1 $, the congruence $ a \equiv g^ \gamma $( $ \mathop{\rm mod}\nolimits \ m $) holds. This $ \gamma $ is called the index of the number $ a $ modulo $ m $ with respect to the basis $ g $ and is denoted by the symbol $ \mathop{\rm ind}\nolimits \ a $ or, more precisely, $ \mathop{\rm ind}\nolimits _{g} \ a $. The properties of an index are broadly the same as the properties of a logarithm. Primitive roots exist only for moduli $ m > 1 $ of the form $ 2,\ 4,\ p^ \alpha ,\ 2p^ \alpha $, where $ p \geq 3 $ is a prime number and $ \alpha \geq 1 $ is an integer. In these cases, the multiplicative group of reduced residue classes modulo $ m $ is simply the cyclic group of order $ \phi (m) $. In the other cases, the groups of reduced residue classes have a much more complex structure.

Many problems in number theory reduce to the question of the solvability or unsolvability of some type of congruence. As a result of this, the theory of congruences, which was first systematically developed by C.F. Gauss (see [5]) and used by him as a foundation of classical number theory, is to this day one of the basic means of solving number-theoretical problems. In this connection, research into the question of the number of solutions of a congruence equation is of fundamental importance to number theory. The simplest types of congruence equations are congruences of the first degree with one unknown $ ax \equiv b $( $ \mathop{\rm mod}\nolimits \ m $). A complete answer to the question of the number of solutions of congruences of the first degree is provided by the following theorem. Let $ (a,\ m) = d $. The congruence $ ax \equiv b $( $ \mathop{\rm mod}\nolimits \ m $) is unsolvable if $ b $ is not divisible by $ d $. When $ b $ is a multiple of $ d $, the congruence has precisely $ d $ solutions.

The question of the solvability of a system of linear congruences

$$ \sum _ { j=1 } ^ s a _{ij} x _{j} \ \equiv \ b _{i} \ ( \mathop{\rm mod}\nolimits \ p),\ \ i = 1 \dots t, $$

modulo a prime number $ p > 2 $ is solved completely in the general theory of linear equations over arbitrary fields. The case of a composite modulus can be reduced to the case of a prime modulus.

The form of algebraic congruences with one unknown which is the next most complex to study is a two-term congruence

$$ x^{n} \ \equiv \ a \ ( \mathop{\rm mod}\nolimits \ m) \ \ \textrm{ when } \ (a,\ m) = 1. $$

If the congruence $ x^{n} \equiv a \ ( \mathop{\rm mod}\nolimits \ m) $ has solutions, then $ a $ is called an $ n $- th power residue modulo $ m $; if it does not, then $ a $ is called an $ n $- th power non-residue modulo $ m $. More specifically, when $ n=2 $, the residues or non-residues are called quadratic, when $ n=3 $, cubic, and when $ n=4 $, bi-quadratic.

The question of the number of solutions of a congruence

$$ f(x) \ \equiv \ 0 \ ( \mathop{\rm mod}\nolimits \ m) $$

modulo a composite number $ m = p _{1} ^ {\alpha _ 1} \dots p _{s} ^ {\alpha _ s} $ reduces to the question of the number of solutions of the congruence

$$ f(x) \ \equiv \ 0 \ ( \mathop{\rm mod}\nolimits \ p _{i} ^ {\alpha _ i} ) $$

for every $ i = 1 \dots s $. The following theorem holds: If $ m _{1} \dots m _{r} $ are pairwise relatively prime (coprime) positive integers, then the number $ N $ of solutions of the congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ m _{1} \dots m _{r} $) can be expressed by the values $ N _{i} $, $ i = 1 \dots r $, which are equal to the number of solutions of the corresponding congruences $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ m _{i} $) according to the formula $ N = \prod _ i=1^{r} N _{i} $. In its turn, the question of the number of solutions of the congruence

$$ f(x) \ \equiv \ 0 \ ( \mathop{\rm mod}\nolimits \ p^ \alpha ),\ \ \alpha > 1 , $$

generally reduces to the question of the number of solutions of the congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $) where $ p $ is a prime number. The following theorem forms the basis of this reduction: Every solution $ x \equiv x _{1} $( $ \mathop{\rm mod}\nolimits \ p $) of a congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $) can be uniquely continued to a solution $ x \equiv x _ \alpha $( $ \mathop{\rm mod}\nolimits \ p^ \alpha $) of the congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p^ \alpha $) provided that the formal derivative $ f ^ {\ \prime} (x _{1} ) \not\equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $), i.e. a unique solution $ x _ \alpha $ of the congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p^ \alpha $) then exists such that $ x _ \alpha \equiv x _{1} $( $ \mathop{\rm mod}\nolimits \ p $).

A similar situation also arises in the case of a congruence equation in several variables, i.e. in non-degenerate cases the question of the number of solutions of the congruence $ F(x _{1} \dots x _{n} ) = 0 $( $ \mathop{\rm mod}\nolimits \ m $), modulo a composite number $ m $ reduces to the question of the number of solutions of the same congruence modulo a prime number $ m $.

An upper bound for the number of solutions of a congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $), where $ f(x) = a _{0} x^{n} + \dots + a _{n} $, $ a _{0} \not\equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $), is given by Lagrange's theorem: The number of solutions of a congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $) does not exceed the degree of the polynomial $ f(x) $. The study of the question of the exact number of solutions of the congruence $ f(x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $) generally meets with considerable difficulties, so that only a few results have so far (1984) been obtained in this area.

The Lagrange theorem ceases to hold in the case of a composite modulus. The above-mentioned specific property of prime numbers is explained by the fact that the residue classes modulo a prime number $ p $ form a field (see Congruence modulo a prime number). Another specific property of prime numbers which is explained by this fact is expressed in the Wilson theorem.

In the study of quadratic congruences

$$ x^{2} \ \equiv \ a \ ( \mathop{\rm mod}\nolimits \ p) $$

modulo a prime number $ p $, as well as in their applications, the Legendre symbol and the Jacobi symbol are introduced.

Congruence equations modulo a prime number in two unknowns (and generally in any number of unknowns),

$$ F(x,\ y) \ \equiv \ 0 \ ( \mathop{\rm mod}\nolimits \ p) , $$

can be treated as equations over the finite prime field of $ p $ elements. Methods of the theory of algebraic functions and algebraic geometry are therefore used in studying them, as well as methods of number theory. Using these methods, an asymptotic expansion in the form

$$ N _{p} \ = \ p + O( \sqrt p ) $$

was first obtained for the number $ N _{p} $ of solutions of a broad class of congruences $ F(x,\ y) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $). The same asymptotic expansions were subsequently deduced using methods of number theory without going beyond the limits of the theory of congruences (see [6]).

Comparatively little is known about algebraic congruences in several unknowns. The following can be considered as a general result (Chevalley's theorem). Let $ F(x _{1} \dots x _{n} ) $ be a polynomial with integer rational coefficients, the degree of which is less than $ n $. The number of solutions of the congruence

$$ F(x _{1} \dots x _{n} ) \ \equiv \ 0 \ ( \mathop{\rm mod}\nolimits \ p) $$

is then divisible by the prime number $ p $. One very strong result for this type of questions was obtained by P. Deligne [9] (see also [10]).

Great value is also attached to research into the question of the number of solutions of congruences in an incomplete residue system. A systematic solution to this type of problems was first attempted by I.M. Vinogradov (see [4]). An example of this type of problems is the problem of the distribution of quadratic residues and non-residues in the set $ 1 \dots p-1 $, which is connected with the study of solutions of the congruence $ y^{2} \equiv x $( $ \mathop{\rm mod}\nolimits \ p $) (see Vinogradov hypotheses; Distribution of power residues and non-residues).

References

[1] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[2] B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian) MR0265267 Zbl 0204.37101
[3] I.M. Vinogradov, "Fundamentals of number theory" , Moscow (1972) (In Russian)
[4] I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) MR0807530 Zbl 0577.01049
[5] C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin) MR0188045 Zbl 21.0166.04
[6] S.A. Stepanov, "A constructive method in the theory of equations over finite fields" Trudy Mat. Inst. Steklov. , 132 (1973) pp. 237–246 (In Russian)
[7] H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) MR0153659 Zbl 1038.11500
[8] K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) MR0249348 Zbl 0169.37502
[9] P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–307 MR0340258 Zbl 0314.14007 Zbl 0287.14001
[10] N.M. Katz, "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields" F.E. Browder (ed.) , Mathematical developments arising from Hilbert problems , Proc. Symp. Pure Math. , 28 , Amer. Math. Soc. (1976) pp. 275–305

Comments

The simultaneous system of congruences $ x \equiv c _{i} $( $ \mathop{\rm mod}\nolimits \ m _{i} $), $ i = 1 \dots r $, where $ ( m _{i} ,\ m _{j} ) = 1 $ for all $ i \neq j $, has a unique residue class modulo $ M = m _{1} \dots m _{r} $ as solution. This is the content of the Chinese remainder theorem.

The procedure to extend solutions of $ f (x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p $) to solutions of $ f (x) \equiv 0 $( $ \mathop{\rm mod}\nolimits \ p^ \alpha $) is described in the Hensel lemma (cf. [a2]), and extends to a technique in the theory of $ p $- adic numbers (cf. $ p $- adic number).

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 5; 7; 8 MR0568909 Zbl 0423.10001
[a2] N. Koblitz, "-adic numbers, -adic analysis and zeta-functions" , Springer (1977) pp. Chapt. 1 MR466081
How to Cite This Entry:
Congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence&oldid=23795
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article