Euler Irregular primes

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(up) Definitions and Notes

[By David Broadhurst, January 2002]

Kummer proved the first case of Fermat's Last Theorem for every prime $p$ that does not divide the numerator of any Bernoulli number $B(2n)$ with $0 \lt 2n \lt p-1.$ Vandiver likewise proved it for Euler-regular primes [Vandiver1940]. (See also

Definition (Vandiver): A prime $p$ is Euler-irregular (E-irregular) if and only if it divides an Euler number $E(2n)$ with $0 \lt 2n \lt p-1.$

Euler numbers are obtained from Taylor coefficients of

$$\frac{1}{\cosh x} = \sum_{k \ge 0} \frac{E(k) x^k}{k!}$$ giving $$E(0) = 1,\, E(2) = -1,\, E(4) =5,\, E(6) = -61,\, E(8) = 1385 ...$$ The smallest E-irregular prime is $p = 19,$ which divides $E(10) = - 50521.$ The first few E-irregular primes are
19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241,
with $p = 241$ dividing both $E(210)$ and $E(238)$ hence having an E-irregularity index of 2.

Vandiver proved [Vandiver1940] that $x^p+y^p= z^p$ has no solution for integers $x,$ $y,$ $z$ with $\gcd(xyz,p) = 1$ if $p$ is Euler-regular. Gut proved [Gut1950] that $x^{2p}+y^{2p}= z^{2p}$ has no solution if $p$ has an E-irregularity index less than 5.

It was proven in [Carlitz1954] that there is an infinity of E-irregular primes. In [Ernvall1975] a stronger result was obtained: there is an infinity of E-irregular primes with residue 1 modulo 8. As in the case of Kummer's B-regular primes, there is as yet no proof that there are infinitely many E-regular primes, though this seems likely to be true.

Like B-irregularity, E-irregularity relates to the divisibility of class numbers of cyclotomic fields [Ernvall1979]. In the case of Kummer's B-irregularity one is working with the trivial character $\chi(k) = 1$ of the Riemann zeta function

$$\zeta(n) = \sum_{k \ge 0} \frac{1}{k^n}$$ which evaluates to $$\zeta(2n) = - (-4\pi^2)^n\frac{B(2n)}{2(2n)!}$$ at the even positive integers. E-irregularity entails the unique character modulo 4, namely $\chi(k)=\sin(k\pi/2)$, with the corresponding Dirichlet series $$\beta(n) = \sum_{k \gt 0}{\frac{\sin(k\pi/2)}{k^n}}$$ evaluating to $$\beta(2n+1) = \pi \left(-\frac{\pi^2}{4}\right)^{\!\!n} \frac{E(2n)}{4(2n)!}$$

at the odd positive integers, with Gregory's formula

$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots$$

being the first example. In [Ernvall1983] it was proven that for every primitive character there is an infinite number of corresponding generalized irregular primes. It is conjectured that the probability of a random prime having E-irregularity index $k$ is given by the Poisson distribution

Conjecture: $P(k) = \displaystyle\frac{e^{-1/2}}{2^k k!}$
and hence that a fraction $1-e^{-1/2}$ (about 39.35%) of primes are E-irregular, as is also conjectured in the case of Kummer's B-irregular primes.

The E-irregular primes less than 10,000 were found in [EM1978], using modular arithmetic to determine divisibility properties of the corresponding Euler numbers. I have re-computed the irregular pairs $(p,2n)$ with $0 \lt 2n \lt p-1 \lt 10,000$ and $E(2n)=0 \pmod{p}.$ The results are tabulated in

The fit to the Poisson conjecture (above) is acceptable:

Index: k = 0 k = 1 k = 2 k = 3 $k \gt 3$
Found: 732 391 86 15 3
Conjectured: 744 372 93 15 2

Samuel Wagstaff has identified a variety of larger E-irregular primes, by factorizing some of the Euler numbers up to E(200). For example, the 278-digit irregular prime


Using ECM (at the p20 level, with 90 runs at B1=11000) to search for titanic cofactors of Euler numbers, I found seven probable primes with between 1000 and 2600 digits:

The first four have been proven prime by Primo; certification of the other three is in progress. There is clearly scope for deeper factoring and hopefully more demanding applications of ECPP.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1E(11848)/7910215 40792 E8 Aug 2022 Euler irregular, ECPP
2E(10168)/1097239206089665 34323 E10 Oct 2023 Euler irregular, ECPP
3 - E(9266)/2129452307358569777 30900 E10 May 2023 Euler irregular, ECPP
4 - E(7894)/19 25790 E10 May 2023 Euler irregular, ECPP
5 - E(7634)/1559 24828 E10 May 2023 Euler irregular, ECPP
6 - E(6658)/85079 21257 c77 Dec 2020 Euler irregular, ECPP
7 - E(5186)/295970922359784619239409649676896529941379763 15954 c63 Mar 2018 Euler irregular, ECPP
8E(3308)/39308792292493140803643373186476368389461245 9516 c8 Jun 2014 Euler irregular, ECPP
9 - E(2762)/2670541 7760 c11 Jul 2004 Euler irregular, ECPP
10E(2220)/392431891068600713525 6011 c8 Feb 2013 Euler irregular, ECPP
11 - E(2202)/53781055550934778283104432814129020709 5938 c8 Feb 2013 Euler irregular, ECPP
12E(2028)/11246153954845684745 5412 c55 Mar 2011 Euler irregular, ECPP
13 - E(1990)/83382\
5258 c8 Feb 2013 Euler irregular, ECPP
14E(1840)/31237282053878368942060412182384934425 4812 c4 May 2011 Euler irregular, ECPP
15E(1736)/13510337079405137518589526468536905 4498 c4 Jan 2004 Euler irregular, ECPP
16 - E(1466)/167900532276654417372106952612534399239 3682 c8 Feb 2013 Euler irregular, ECPP
3671 c4 Dec 2003 Euler irregular, ECPP
18 - E(1174)/50550511342697072710795058639332351763 2829 c8 Feb 2013 Euler irregular, ECPP
19 - E(1142)/6233437695283865492412648122\
2697 c77 Apr 2015 Euler irregular, ECPP
20 - E(1078)/361898544439043 2578 c4 Feb 2002 Euler irregular, ECPP

(up) References

L. Carlitz, "Note on irregular primes," Proc. Amer. Math. Soc., 5 (1954) 329--331.  MR 15,778b
R. Ernvall and T. Metsänkylä, "Cyclotomic invariants and e-irregular primes," Math. Comp., 32 (1978) 617--629.  MR 80c:12004a
R. Ernvall, "On the distribution mod 8 of the E-irregular primes," Ann. Acad. Sci. Fenn. Ser. A, 1 (1975) 195--198.  MR 52:5594
R. Ernvall, "Generalized Bernoulli numbers, generalized irregular primes and class number," Ann. Univ. Turku. Ser. A, 1:178 (1979) 72 pp..  MR 80m:12002
R. Ernvall, "Generalized irregular primes," Mathematika, 30:1 (1983) 67--73.  MR 85g:11022
M. Gut, "Eulersche zahlen und grosser Fermat'sche satz," Comment. Math. Helv., 24 (1950) 73--99.  MR 12,243d
H. S. Vandiver, "Note on Euler number criteria for the first case of Fermat's last theorem," Amer. J. Math., 62 (1940) 79--82.  MR 1,200d
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