View source for Irregular prime number

An odd [[Prime number|prime number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526701.png" /> for which the number of classes of ideals in the [[Cyclotomic field|cyclotomic field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526702.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526703.png" />. All other odd prime numbers are called regular.

Kummer's test allows one to solve for each given prime number the problem of whether it is regular or not: For an odd prime number to be regular it is necessary and sufficient that none of the numerators of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526704.png" /> [[Bernoulli numbers|Bernoulli numbers]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526705.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526706.png" /> (cf. [[#References|[1]]]).

The problem of the distribution of regular and irregular prime numbers arose in this connection. Tables of the Bernoulli numbers and Kummer's test indicated that among the first hundred there are only three irregular prime numbers, 37, 59, 67 (the numerators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i0526709.png" /> are multiples of 37, 59 and 67, respectively). E. Kummer conjectured that there are on the average twice as many regular prime numbers as irregular ones. Later C.L. Siegel [[#References|[2]]] conjectured that the ratio of irregular prime numbers to regular prime numbers contained in an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267010.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267012.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267013.png" /> is the base of natural logarithms). Up till now (1989) it is only known that the number of irregular prime numbers is infinite. There are 439 regular and 285 irregular prime numbers among the odd numbers smaller than 5500, cf. [[#References|[3]]].

For any regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267014.png" /> the Fermat equation

<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/i/i052/i052670/i05267015.png" /></td> </tr></table>

does not have non-zero solutions in the rational numbers [[#References|[1]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267016.png" /> be an irregular prime number, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267017.png" /> be the indices of the Bernoulli numbers among <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267018.png" /> whose numerators are divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267019.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267021.png" /> be natural numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267022.png" /> is a prime number smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267024.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267025.png" />. Let

<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/i/i052/i052670/i05267026.png" /></td> </tr></table>

<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/i/i052/i052670/i05267027.png" /></td> </tr></table>

If for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267029.png" />,

<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/i/i052/i052670/i05267030.png" /></td> </tr></table>

then for the irregular prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267031.png" /> Fermat's theorem holds, i.e. the Fermat equation is unsolvable in the non-zero rational numbers. This is called Vandiver's test. The truth of Fermat's theorem for all exponents smaller than 5500 has been proved by using Vandiver's test (cf. [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kummer,   "Allgemeiner Beweis des Fermat'schen Satzes, dass die Gleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267032.png" /> durch ganze Zahlen unlösbar ist, für alle diejenigen Potentz-Exponenten <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267033.png" />, welche ungerade Primzahlen sind und in den Zählern der ersten <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267034.png" /> Bernoulli'schen Zahlen als Factoren nicht Vorkommen"  ''J. Reine Angew. Math.'' , '''40'''  (1850)  pp. 130–138</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.L. Siegel,   "Zu zwei Bemerkungen Kummers"  ''Nachr. Akad. Wiss. Göttingen Math. Phys. Kl.'' , '''6'''  (1964)  pp. 51–57</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[4]</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</TD></TR></table>



====Comments====
The truth of Fermat's theorem has been established for all exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267035.png" /> by S. Wagstaff [[#References|[a1]]].

His computations show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267036.png" /> of the 11733 odd prime numbers smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267037.png" /> are regular. This is in close agreement with Siegel's conjecture, which expects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267038.png" /> of all prime numbers to be regular.

More generally, one defines the index of irregularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267039.png" /> of an odd prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267040.png" /> as the number of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267041.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267042.png" /> divides the numerator of the Bernoulli number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267043.png" />. The regular prime numbers are the prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267044.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267045.png" />. Heuristically, one expects the fraction of prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267046.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267047.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267048.png" />, and this is confirmed by the data in [[#References|[a1]]]. It was proved by Eichler that the first case of Fermat's theorem holds for a prime exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267049.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052670/i05267050.png" /> (cf. [[#References|[a2]]]). See also [[Fermat great theorem|Fermat great theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Wagstaff,   "The irregular primes to 125,000"  ''Math. Comp.'' , '''32'''  (1978)  pp. 583–591</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.C. Washington,   "Introduction to cyclotomic fields" , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  S. Lang,   "Cyclotomic fields" , '''1–2''' , Springer  (1978–1980)</TD></TR></table>