Irregular Primes

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

This page is about one of those forms.

(up) Definitions and Notes

[By David Broadhurst, 29 November 2001]

In 1847 Gabriel Lame (1795-1870) gave a talk to l'Academie des Sciences in Paris, in which he claimed to have proved Fermat's last theorem. Joseph Liouville (1809-1882) rose to his feet to object. He pointed out that Lame had wrongly assumed unique factorization in the ring of $p$-cyclotomic integers, generated by $\exp(2\pi i/p)$ with prime $p$.

Yet from this contretemps came something of real value. Ernst Eduard Kummer (1810-1893) had already studied the failure of unique factorization in cyclotomic fields and soon formulated a theory of ideals, which was later developed by Julius Wilhelm Richard Dedekind (1831-1916). Kummer was able to prove Fermat's last theorem for all prime exponents that are "regular".

A prime $p$ is irregular if and only if $p$ divides the class number of the cyclotomic field generated by $\exp(2\pi i/p).$ A regular prime is one that is not irregular.

Equivalently, but more conveniently, an odd prime $p$ is irregular if and only if $p$ divides the numerator of a Bernoulli number $B(2n)$ with $2n + 1 \lt p.$

Bernoulli numbers are obtained from Taylor coefficients of $$\frac{x}{\exp(x)-1} = \sum_{k \ge 0}{\frac{B(k) x^k}{k!}}$$

from which it follows that $B(2n+1)=0$ for $n\gt0.$

The first Bernoulli number $B(2n)$ with a non-trivial numerator greater than $n$ is B(12) = -691/2730, from which it follows that 691 is an irregular prime. The smallest irregular prime is 37, which divides the numerator of B(32) =  -7709321041217/510.

By analyzing all the Bernoulli numbers up to $B(2n),$ one may find all the irregular primes up to $2n+3.$

By 1874, Kummer was able to prove Fermat's last theorem for all odd prime exponents less than 165, save for the eight irregular primes

p = 37, 59, 67, 101, 103, 131, 149, 157
which divide the numerators of $B(n),$ with
$n$ = 32, 44, 58, 68, 24, 22, 130, 62
respectively. Moreover 157 divides the numerator of B(110), as well as that of B(62), and hence is assigned an irregularity index of 2. Note that to determine divisibility we need not compute the large numerators of these Bernoulli numbers; modular arithmetic suffices.

In 1857, Kummer was awarded a 3000 FF prize, in a competition for which he had not entered. L'Academie des Sciences had been holding this prize in reserve, hoping for a full proof of Fermat's last theorem. One may surmise that Kummer's analysis showed them that their challenge was too hard and that in consequence they chose his work as the next best thing.

Kummer hoped that the number of irregular primes might be finite.  This was disproved by Jensen, in 1915. A simple proof that the irregular primes are infinite in number, by contradiction of the converse, was given in [Carlitz1954].  Ironically, it is not known whether the number of regular primes is infinite, though there is both an heuristic argument [Siegel1964] and also extensive empirical evidence [Wagstaff78] to support the conjecture that regular primes comprise an asymptotic proportion e-1/2, or about 60.65% of all primes, with about 39.35% being consequently irregular.

Irregular primes have been enumerated, up to increasing limits, in [Johnson1974], [Johnson1975], [Wagstaff78], [TW1987], [BCS1992], [BCEM1993], [BCEMS2000], with increasing efficiency of the modular arithmetic. The most recent of these works determined all the irregular primes less than 12,000,000.

Samuel Wagstaff has identified a variety of larger irregular primes, by factorizing some of the Bernoulli numbers up to B(300). For example, one finds in http://www.cerias.purdue.edu/homes/ssw/bernoulli/enum the 322-digit irregular prime

2*3*B(298)/(149*26113*223529*371472263795653589766634977803).

Recently, I began to look for titanic irregular primes, with at least 1000 decimal digits, and soon found the following 8 instances:

2*3*B(674)/337
-2*3*5*17*173*B(688)/(43*33617*1323797*465776197109)
2*3*B(734)/(157*367*6547*75401119*170508089)
2*3*59*B(754)/(13*29*883462452530494157)
2*3*23*B(814)/(11*37*69803353*335055893*351085907*520460183*30348030379*17043083582983)
-2*3*5*17*107*B(848)/(53*1789*4519*5051145078213134269)
-2*3*5*1229*B(1228)/(307*3399696012787)
-2*3*5*7*13*619*1237*B(1236)/(103*939551962476779*157517441360851951)
all of which were proven prime by Marcel Martin's Primo.  The largest has 2276 decimal digits.

Bouk de Water and I have continued ECM factorization attempts on the numerators of Bernoulli numbers up to B(2000), finding the titanic irregular prime

-2*3*5*23*B(748)/(11*17*3853*138192830377045750339532383)
and the following four probable primes,
2*3*107*B(1802)/(17*53*17448089)
-2*3*5*7*13*19*37*103*109*307*613*919*B(1836)/(17*3517*3927024469727)
2*3*11*23*1871*B(1870)/(5*17)
-2*3*5*7*13*B(1884)/(157*3697*173981)
The smallest of these PrPs has 3641 digits; the largest has 3844 digits. All four are feasible targets for proof ECPP.

I thank Richard Crandall for advice. Historical information was gleaned from a wide variety of websites. Paulo Ribenboim's excellent account in "The new book of prime number records" [Ribenboim95] provided most of the appended references. I have been able to read all but the second and last of these nine papers.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1 - 30 · Bern(10264)/262578313564364605963 28506 c94 May 2021 Irregular, ECPP
2546351925018076058 · Bern(9702)/129255048976106804786904258880518941 26709 c77 Jan 2021 Irregular, ECPP
3798 · Bern(8766)/14670751334144820770719 23743 c94 Jan 2021 Irregular, ECPP
4348054 · Bern(8286)/1570865077944473903275073668721 22234 E1 Dec 2022 Irregular, ECPP
56 · Bern(5534)/226840561549600012633271691723599339 13862 c71 Feb 2014 Irregular, ECPP
64410546 · Bern(5526)/9712202742835546740714595866405369616019 13840 c63 Apr 2018 Irregular, ECPP
76 · Bern(5462)/23238026668982614152809832227 13657 c64 May 2013 Irregular, ECPP
86 · Bern(5078)/643283455240626084534218914061 12533 c63 May 2013 Irregular, ECPP
91258566 · Bern(4462)/6610083971965402783802518108033 10763 c64 Mar 2013 Irregular, ECPP
106 · Bern(4306)/2153 10342 FE8 Apr 2009 Irregular, ECPP
116 · Bern(3458)/28329084584758278770932715893606309 7945 c8 Feb 2013 Irregular, ECPP
12 - 30 · Bern(3176)/668969310005687298938683\
37398130897205591897362591275370617658634396391181		 7138 c63 Nov 2016 Irregular, ECPP
13 - 10365630 · Bern(3100)/16703661161128644816\
99585217650438278080436881373643007997602585219667		 6943 c63 Sep 2016 Irregular ECPP
146 · Bern(2974)/19622040971147542470479091157507 6637 c8 Feb 2013 Irregular, ECPP
15274386 · Bern(2622)/8518594882415401157891061256276973722693 5701 c8 Feb 2013 Irregular, ECPP
16 - 30 · Bern(2504)/1248230090315232335602406\
37343822165241758149026675581438903418303340323897		 5354 c63 Feb 2013 Irregular ECPP
1733957462 · Bern(2370)/40685 5083 c11 Jun 2003 Irregular, ECPP
18276474 · Bern(2030)/469951697500688159155 4200 c8 Dec 2003 Irregular, ECPP
19 - 2730 · Bern(1884)/100983617849 3844 c8 Nov 2003 Irregular, ECPP
202840178 · Bern(1870)/85 3821 c8 Dec 2003 Irregular, ECPP

(up) References

BCEM1993
J. Buhler, R. Crandall, R. Ernvall and T. Metsänkylä, "Irregular primes and cyclotomic invariants to four million," Math. Comp., 61:203 (1993) 151--153.  MR 93k:11014
BCEMS2000
J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million," J. Symbolic Comput., 31:1--2 (2001) 89--96.  MR 2001m:11220
BCS1992
J. P. Buhler, R. E. Crandall and R. W. Sompolski, "Irregular primes to one million," Math. Comp., 59:200 (1992) 717--722.  MR 93a:11106
BH2011
Buhler, J. P. and Harvey, D., "Irregular primes to 163 million," Math. Comp., 80:276 (2011) 2435--2444.  (http://dx.doi.org/10.1090/S0025-5718-2011-02461-0) MR 2813369
Carlitz1954
L. Carlitz, "Note on irregular primes," Proc. Amer. Math. Soc., 5 (1954) 329--331.  MR 15,778b
Johnson1974
W. Johnson, "Irregular prime divisors of the bernoulli numbers," Math. Comp., 28 (1974) 653--657.  MR 50:229
Johnson1975
W. Johnson, "Irregular primes and cyclotomic invariants," Math. Comp., 29 (1975) 113--120.  MR 51:12781
Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Siegel1964
C. L. Siegel, "Zu zwei Bemerkungken Kummers," Nachr. Akad. d. Wiss. Goettingen, Math. Phys. KI., II (1964) 51--62.
TW1987
J. W. Tanner and S. S. Wagstaff Jr., "New congruences for the Bernoulli numbers," Math. Comp., 48 (1987) 341--350.  MR 87m:11017
Wagstaff78
Wagstaff, Jr., S. S., "The irregular primes to 125,000," Math. Comp., 32 (1978) 583-591.  MR 58:10711 [Kummer was able to show that FLT was true for the regular primes.]
Washington82
L. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics Vol, 83, Springer-Verlag, 1982.  New York, NY, pp. xi+389, ISBN 0-387-90622-3. (There is a later edition).  MR 85g:11001
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