Difference between revisions of "Fermat great theorem"
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''Fermat's famous theorem, Fermat's big theorem, Fermat's last theorem'' | ''Fermat's famous theorem, Fermat's big theorem, Fermat's last theorem'' | ||
− | The assertion that for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383901.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383902.png" /> (the Fermat equation) has no solution in non-zero integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383903.png" />. It was stated by P. Fermat in about 1630 in the margins of his copy of the book Aritmetika [[#References|[1]]] by Diophantus as follows: | + | The assertion that for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383901.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383902.png" /> (the Fermat equation) has no solution in non-zero integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383903.png" />. It was stated by P. Fermat in about 1630 in the margins of his copy of the book Aritmetika [[#References|[1]]] by Diophantus as follows: "It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent" . And he then added: "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain" . A proof of the theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f0383904.png" /> was found in Fermat's papers. No general proof has so far been obtained (1984), despite the efforts of many mathematicians (both professional and amateur). An unhealthy interest in proving this theorem was stimulated at one time by a large international prize, which was abolished at the end of the First World War. |
It has been conjectured that there is no proof of Fermat's last theorem at all. | It has been conjectured that there is no proof of Fermat's last theorem at all. | ||
Line 42: | Line 42: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> Diophantus of Alexandria, "Aritmetika and the book on polygonal numbers" , Moscow (1974) (In Russian; translated from Greek)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H.M. Edwards, "Fermat's last theorem. A genetic introduction to algebraic number theory" , Springer (1977)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> E. Kummer, "Bestimmung der Anzahl nicht äquivalenter Classen für die aus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390147.png" />ten Wurzeln der Einheit gebildeten complexen Zahlen" ''J. Reine Angew. Math.'' , '''40''' (1850) pp. 93–116</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> E. Kummer, "Zwei besondere Untersuchungen über die Classen-Anzahl und über die Einheiten der aus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390148.png" />ten Wurzeln der Einheit gebildeten complexen Zahlen" ''J. Reine Angew. Math.'' , '''40''' (1850) pp. 117–129</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> E. Krummer, "Algemeiner Beweis des Fermat'schen Satzes, dass die Gleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390149.png" /> durch ganze Zahlen unlösbar ist, für diejenige Potenz-Exponenten <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390150.png" />, welche ungerade Primzahlen sind und in den Zählern der ersten <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390151.png" /> Bernoullischen Zahlen als Faktoren nicht vorkommen" ''J. Reine Angew. Math.'' , '''40''' (1950) pp. 130–138</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top"> E. Kummer, "Einige Sätze über die aus den Wurzeln der Gleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390152.png" /> gebildeten complexen Zahlen, für den Fall, dass die Klassenanzahl durch <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390153.png" /> teilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes" ''Abh. Akad. Wiss. Berlin, Math. Kl.'' (1857) pp. 41–74</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Vandiver, "Fermat's last theorem" ''Amer. Math. Monthly'' , '''53''' (1946) pp. 555–578 {{MR|}} {{ZBL|52.0161.13}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> P. Ribenboim, "Thirteen lectures on Fermat's last theorem" , Springer (1979)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M. Krasner, "Sur le premier cas du théorème de Fermat" ''C.R. Acad. Sci. Paris'' , '''199''' (1934) pp. 256–258 {{MR|}} {{ZBL|0010.00702}} {{ZBL|60.0129.01}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Brückner, "Zum Beweis des ersten Falles der Fermatschen Vermutung für pseudo requlären Primzahlen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390154.png" />" ''J. Reine Angew. Math.'' , '''253''' (1972) pp. 15–18</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P.N. Remorov, "On Kummer's theorem" ''Uchen. Zap. Leningrad. Gosudarstv. Univ. Ser. Mat. Nauk'' , '''144''' : 23 (1952) pp. 26–34 (In Russian) {{MR|81310}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> M. Eichler, "Eine Bemerkung zur Fermatschen Vermutung" ''Acta Arith.'' , '''11''' (1965) pp. 129–131 {{MR|0182607}} {{ZBL|0135.09401}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> H. Vandiver, "Fermat's last theorem and the second factor in the cyclotomic class number" ''Bull. Amer. Math. Soc.'' , '''40''' (1934) pp. 118–126 {{MR|1562807}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> H. Vandiver, "On Fermat's last theorem" ''Trans. Amer. Math. Soc.'' , '''31''' (1929) pp. 613–642 {{MR|1501503}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> H.S. Vandiver, "Examination of methods of attack on the second case of Fermat's last theorem" ''Proc. Nat. Acad. Sci. USA'' , '''40''' : 8 (1954) pp. 732–735 {{MR|62758}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> S. Wagstaff, "The irregular primes to 125.000" ''Math. Comp.'' , '''32''' (1978) pp. 583–591 {{MR|491465}} {{ZBL|}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> A. Wieferich, "Zum letzter Fermatschen Theorem" ''J. Reine Angew. Math.'' , '''136''' (1909) pp. 293–302</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> D. Mirimanoff, "Zum letzter Fermatschen Theorem" ''J. Reine Angew. Math.'' , '''139''' (1911) pp. 309–324</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> D.H. Lehmer, "On Fermat's quotient, base 2" ''Math. Comp.'' , '''36''' (1981) pp. 289–290 {{MR|595064}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> P. Furtwängler, "Letzter Fermat'scher Satz und Eisenstein'sches Reziprozitätsprinzip" ''Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl. IIa'' , '''121''' (1912) pp. 589–592</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> G. Terjanian, "Sur l'equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038390/f038390155.png" />" ''C.R. Acad. Sci. Paris'' , '''A285-B285''' : 16 (1977) pp. 973A-975A (English abstract) {{MR|0498370}} {{ZBL|}} </TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> K. Inkeri, "Abschätzungen für eventuelle Lösungen der Gleichung im Fermatschen problem" ''Ann. Univ. Turku Ser. A'' , '''16''' : 1 (1953) pp. 3–9 {{MR|0058629}} {{ZBL|0051.28003}} </TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top"> M.M. Postnikov, "An introduction to algebraic number theory" , Moscow (1982) (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top"> G. Faltings, "Endichkeitssätze für abelsche Varietäten über Zahlkörpern" ''Invent. Math.'' , '''73''' (1983) pp. 349–366</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top"> D.R. Heath-Brown, "Fermat's last theorem for "almost all" exponents" ''Bull. London Math. Soc.'' , '''17''' (1985) pp. 15–16</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top"> L.M. Adleman, D.R. Heath-Brown, "The first case of Fermat's last theorem" ''Invent. Math.'' , '''79''' (1985) pp. 409–416 {{MR|778135}} {{ZBL|}} </TD></TR></table> |
Line 54: | Line 54: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Wagon, "Fermat's last theorem" ''Math. Intelligencer'' , '''8''' : 1 (1986) pp. 59–61 {{MR|823221}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Ribenboim, "Recent results about Fermat's last theorem" ''Cand. Math. Bull.'' , '''20''' (1977) pp. 229–242 {{MR|463088}} {{ZBL|}} </TD></TR></table> |
Revision as of 21:52, 30 March 2012
Fermat's famous theorem, Fermat's big theorem, Fermat's last theorem
The assertion that for any natural number the equation
(the Fermat equation) has no solution in non-zero integers
. It was stated by P. Fermat in about 1630 in the margins of his copy of the book Aritmetika [1] by Diophantus as follows: "It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent" . And he then added: "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain" . A proof of the theorem for
was found in Fermat's papers. No general proof has so far been obtained (1984), despite the efforts of many mathematicians (both professional and amateur). An unhealthy interest in proving this theorem was stimulated at one time by a large international prize, which was abolished at the end of the First World War.
It has been conjectured that there is no proof of Fermat's last theorem at all.
For the theorem was proved by L. Euler (1770), for
by P.G.L. Dirichlet and A. Legendre (1825), and for
by G. Lamé (1839) (see [2]). It is sufficient to prove the theorem for
and for every prime exponent
, that is, it is enough to prove that the equation
![]() | (1) |
has no solutions in non-zero relatively-prime integers . One can also assume that
and
are relatively prime to
. For a proof of Fermat's theorem one considers two cases: case 1 when
, and case 2 when
. The proof of the second case is more difficult and is usually carried out by the method of infinite descent. An important contribution to proving Fermat's last theorem was made by E. Kummer, who invented a fundamentally new method based on his development of the arithmetic theory of a cyclotomic field. It makes use of the fact that in the field
,
, the left-hand side of equation (1) splits into linear factors
, which are
-th powers of ideal numbers (cf. Ideal number) in
in case 1 and differ from
-th powers by a factor
,
, in case 2. If
divides the numerators of the Bernoulli numbers
(
), then by the regularity criterion
does not divide the class number
of
and these ideal numbers are principal. Kummer
proved Fermat's theorem in this case. It is not known whether the number of regular numbers is infinite or finite (by Jensen's theorem the number of irregular prime numbers is infinite [4]). Kummer [5] proved the theorem for some irregular prime numbers and also established its validity for all
. In case 1 he showed that (1) implies the congruences
![]() |
![]() |
which are valid for any permutation of . Hence he obtained that if equation (1) has a solution in case 1, then for
,
![]() | (2) |
In case 2 Kummer proved Fermat's theorem under the following conditions: 1) ,
, where
is the first factor of the class number of
(this is equivalent to the requirement that only one of the numerators of the numbers
, where
, is divisible by
); 2)
(
); and 3) there is an ideal modulo which the unit
![]() |
is not congruent to the -th power of an integer in
, where
is a primitive root modulo
and
![]() |
Kummer's method has been widely developed in several articles on Fermat's last theorem (see [6], [7]). It has been established that (2) holds if (1) does in case 1 for , 9, 11, 13, 15, 17, 19. Under the same conditions M. Krasner [8] showed that there is a number
such that for
(2) is true for all numbers
, where
.
H. Brückner [9] showed that the amount of numbers ,
, with numerators divisible by
is greater than
. Suppose that
,
. P. Remorov [10] showed that there are constants
and
,
, such that for all
,
, case 1 of the Fermat theorem is true. M. Eichler [11] established that case 1 is true for
, where
is the index of irregularity of
,
. H. Vandiver [12] proved case 1 for
, where
is the second factor of the class number of
. He obtained interesting results on case 2 in [13] and [6]. For example, he showed that the Fermat theorem is true under the following conditions: 1)
; and 2)
(
),
. The following theorem is most important: Let
be an irregular prime number and let
be the indices of the Bernoulli numbers among
with numerators divisible by
; if none of the units
(
) is congruent to the
-th power of an integer in
modulo
, where
is the prime ideal dividing a prime number
with
(
), then Fermat's theorem is true. From this Vandiver [14] obtained an effectively-verifiable criterion for irregular prime numbers by means of which the Fermat theorem has been proved on a computer for all
(see [15]).
There are various results on case 1 of Fermat's last theorem. As early as 1823 Legendre published a result of S. Germain: If there is a prime number such that the congruence
(
) has no integer solutions
not divisible by
, and
is not a
-th power residue modulo
, then case 1 of the Fermat theorem holds (see [2]). Hence he showed that if at least one of the numbers
,
(
),
, is prime, then case 1 holds. This proposition has been extended to all
. A. Wieferich [16] discovered the following criterion: If
, where
is the Fermat quotient, then case 1 is true. D. Mirimanoff [17] proved this for
. Subsequently, case 1 was established by a number of other authors for all
for which
, where
is any prime number
. From this the first case of Fermat's theorem follows for
, where
contain only prime numbers
in their prime factorizations. Calculations on a computer showed [18] that among the numbers
only two:
and
satisfy the condition
, but for these
. This proves case 1 for all
. P. Furtwängler [19] gave fairly simple new proofs of the results of Wieferich and Mirimanoff based on Eisenstein's reciprocity law. He also proved that if
is a solution of (1) and
, then
, where
but
, or
but
, or
but
.
A great variety of other criteria are known for case 1 of the Fermat theorem. They are connected with the solvability of certain congruences or with the existence of prime numbers of a certain form. The equation is not valid if
divides neither
nor
(see [20]). It is impossible in practice to produce a counterexample to Fermat's last theorem. K. Inkeri [21] showed that if the integers
,
, satisfy (1), then
, and in case 1:
.
Fermat's last theorem can be stated as follows: For every natural number there are no rational points on the Fermat curve
except the trivial ones,
and
. Rational points on the Fermat curve have been studied by methods of algebraic geometry. By these methods it has been proved (1983) that the number of rational points on the Fermat curve is finite in every case. This follows from the Mordell conjecture, which was proved by G. Faltings [23]. D.R. Heath-Brown has shown, using the Mordell conjecture, that Fermat's last theorem holds for almost-all primes
, cf. [24]. Also, by methods of analytic number theory, L.M. Adleman, Foury and Heath-Brown have shown that case 1 holds for infinitely many primes
, cf. [25].
One can look at Fermat's equation in algebraic integers, entire functions, matrices, etc. There is a generalization of Fermat's theorem for equations of the form .
References
[1] | Diophantus of Alexandria, "Aritmetika and the book on polygonal numbers" , Moscow (1974) (In Russian; translated from Greek) |
[2] | H.M. Edwards, "Fermat's last theorem. A genetic introduction to algebraic number theory" , Springer (1977) |
[3a] | E. Kummer, "Bestimmung der Anzahl nicht äquivalenter Classen für die aus ![]() |
[3b] | E. Kummer, "Zwei besondere Untersuchungen über die Classen-Anzahl und über die Einheiten der aus ![]() |
[3c] | E. Krummer, "Algemeiner Beweis des Fermat'schen Satzes, dass die Gleichung ![]() ![]() ![]() |
[4] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902 |
[5] | E. Kummer, "Einige Sätze über die aus den Wurzeln der Gleichung ![]() ![]() |
[6] | H. Vandiver, "Fermat's last theorem" Amer. Math. Monthly , 53 (1946) pp. 555–578 Zbl 52.0161.13 |
[7] | P. Ribenboim, "Thirteen lectures on Fermat's last theorem" , Springer (1979) |
[8] | M. Krasner, "Sur le premier cas du théorème de Fermat" C.R. Acad. Sci. Paris , 199 (1934) pp. 256–258 Zbl 0010.00702 Zbl 60.0129.01 |
[9] | H. Brückner, "Zum Beweis des ersten Falles der Fermatschen Vermutung für pseudo requlären Primzahlen ![]() |
[10] | P.N. Remorov, "On Kummer's theorem" Uchen. Zap. Leningrad. Gosudarstv. Univ. Ser. Mat. Nauk , 144 : 23 (1952) pp. 26–34 (In Russian) MR81310 |
[11] | M. Eichler, "Eine Bemerkung zur Fermatschen Vermutung" Acta Arith. , 11 (1965) pp. 129–131 MR0182607 Zbl 0135.09401 |
[12] | H. Vandiver, "Fermat's last theorem and the second factor in the cyclotomic class number" Bull. Amer. Math. Soc. , 40 (1934) pp. 118–126 MR1562807 |
[13] | H. Vandiver, "On Fermat's last theorem" Trans. Amer. Math. Soc. , 31 (1929) pp. 613–642 MR1501503 |
[14] | H.S. Vandiver, "Examination of methods of attack on the second case of Fermat's last theorem" Proc. Nat. Acad. Sci. USA , 40 : 8 (1954) pp. 732–735 MR62758 |
[15] | S. Wagstaff, "The irregular primes to 125.000" Math. Comp. , 32 (1978) pp. 583–591 MR491465 |
[16] | A. Wieferich, "Zum letzter Fermatschen Theorem" J. Reine Angew. Math. , 136 (1909) pp. 293–302 |
[17] | D. Mirimanoff, "Zum letzter Fermatschen Theorem" J. Reine Angew. Math. , 139 (1911) pp. 309–324 |
[18] | D.H. Lehmer, "On Fermat's quotient, base 2" Math. Comp. , 36 (1981) pp. 289–290 MR595064 |
[19] | P. Furtwängler, "Letzter Fermat'scher Satz und Eisenstein'sches Reziprozitätsprinzip" Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl. IIa , 121 (1912) pp. 589–592 |
[20] | G. Terjanian, "Sur l'equation ![]() |
[21] | K. Inkeri, "Abschätzungen für eventuelle Lösungen der Gleichung im Fermatschen problem" Ann. Univ. Turku Ser. A , 16 : 1 (1953) pp. 3–9 MR0058629 Zbl 0051.28003 |
[22] | M.M. Postnikov, "An introduction to algebraic number theory" , Moscow (1982) (In Russian) |
[23] | G. Faltings, "Endichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 |
[24] | D.R. Heath-Brown, "Fermat's last theorem for "almost all" exponents" Bull. London Math. Soc. , 17 (1985) pp. 15–16 |
[25] | L.M. Adleman, D.R. Heath-Brown, "The first case of Fermat's last theorem" Invent. Math. , 79 (1985) pp. 409–416 MR778135 |
Comments
In fact, Heath-Brown and, independently, A. Granville, cf. [a1], have proved that the density of the exponents for which Fermat's last theorem holds is one.
It is now (1988) known that Fermat's last theorem holds for all , and that case 1 holds for all primes up to
, cf. [a2].
Recently (1987), K. Ribet, using ideas of G. Frey and J.-P. Serre, showed that Fermat's last theorem is implied by the Weil–Taniyama conjecture in the theory of elliptic curves (cf. Elliptic curve).
References
[a1] | S. Wagon, "Fermat's last theorem" Math. Intelligencer , 8 : 1 (1986) pp. 59–61 MR823221 |
[a2] | P. Ribenboim, "Recent results about Fermat's last theorem" Cand. Math. Bull. , 20 (1977) pp. 229–242 MR463088 |
Fermat great theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fermat_great_theorem&oldid=23829