## Irregular Primes |

[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

which divide the numerators of $B(n),$ withp= 37, 59, 67, 101, 103, 131, 149, 157

$n$ = 32, 44, 58, 68, 24, 22, 130, 62respectively. 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)/337all of which were proven prime by Marcel Martin's Primo. The largest has 2276 decimal digits.

-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)

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)The smallest of these PrPs has 3641 digits; the largest has 3844 digits. All four are feasible targets for proof ECPP.

-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)

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.

rank prime digits who when comment 1 - 30 · Bern(10264)/(1040513 · 252354668864651)28506 c94 May 2021 Irregular, ECPP 2 546351925018076058 · Bern(9702)/12925504897610680478690425888051894126709 c77 Jan 2021 Irregular, ECPP 3 798 · Bern(8766)/(2267959 · 6468702182951641)23743 c94 Jan 2021 Irregular, ECPP 4 6 · Bern(5534)/(89651360098907 · 22027790155387 · 114866371)13862 c71 Feb 2014 Irregular, ECPP 5 4410546 · Bern(5526)/(4931516285027 · 1969415121333695957254369297)13840 c63 Apr 2018 Irregular, ECPP 6 6 · Bern(5462)/(724389557 · 8572589 · 3742097186099)13657 c64 May 2013 Irregular, ECPP 7 6 · Bern(5078)/(64424527603 · 9985070580644364287)12533 c63 May 2013 Irregular, ECPP 8 1258566 · Bern(4462)/(2231 · 596141126178107 · 4970022131749)10763 c64 Mar 2013 Irregular, ECPP 9 6 · Bern(4306)/215310342 FE8 Apr 2009 Irregular, ECPP 10 6 · Bern(3458)/283290845847582787709327158936063097945 c8 Feb 2013 Irregular, ECPP 11 - 30 · Bern(3176)/(169908471493279 · 905130251538800883547330531 · 434990809309147283469396721753169)7138 c63 Nov 2016 Irregular, ECPP 12 - 10365630 · Bern(3100)/(140592076277 · 66260150981141825531862457 · 179307479508256366206520177467103)6943 c63 Sep 2016 Irregular ECPP 13 6 · Bern(2974)/196220409711475424704790911575076637 c8 Feb 2013 Irregular, ECPP 14 274386 · Bern(2622)/85185948824154011578910612562769737226935701 c8 Feb 2013 Irregular, ECPP 15 - 30 · Bern(2504)/(313 · 424524649821233650433 · 117180678030577350578887 · 8016621720796146291948744439)5354 c63 Feb 2013 Irregular ECPP 16 33957462 · Bern(2370)/406855083 c11 Jun 2003 Irregular, ECPP 17 276474 · Bern(2030)/(19426085 · 24191786327543)4200 c8 Dec 2003 Irregular, ECPP 18 - 2730 · Bern(1884)/1009836178493844 c8 Nov 2003 Irregular, ECPP 19 2840178 · Bern(1870)/853821 c8 Dec 2003 Irregular, ECPP 20 - 197676570 · 18851280661 · Bern(1836)/(59789 · 3927024469727)3734 c8 Nov 2003 Irregular, ECPP

- The Top 20: Euler Irregular Primes

- BCEM1993
J. Buhler,R. Crandall,R. ErnvallandT. 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. MetsankylaandM. 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. CrandallandR. W. Sompolski, "Irregular primes to one million,"Math. Comp.,59:200 (1992) 717--722.MR 93a:11106- BH2011
Buhler, J. P.andHarvey, 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. TannerandS. 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

Chris K. Caldwell
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