## Euler Irregular primes |

[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 http://groups.yahoo.com/group/primenumbers/message/4197.)

Definition(Vandiver): A prime $p$ isEuler-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 are19, 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

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.Conjecture:$P(k) = \displaystyle\frac{e^{-1/2}}{2^k k!}$

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 http://groups.yahoo.com/group/primeform/files/Irreg/euler.txt

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

Index: k= 0k= 1k= 2k= 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

-E(194)/(34110029*28024555486506389*2436437750204310804841)

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:

-E(510)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.

-E(638)/(7235862947323*11411779188663863*526900327479624797)

-E(886)/(149*461)

-E(902)/(9756496279*314344516832998594237)

E(1004)/(5*541*214363*80533376783)

E(1028)/(5*1283*56837916301577)

-E(1078)/(71*433*11771738101)

rank prime digits who when comment 1 E(3308)/393087922924931408036433731864763683894612459516 c8 Jun 2014 Euler irregular, ECPP 2 - E(2762)/26705417760 c11 Jul 2004 Euler irregular, ECPP 3 E(2220)/3924318910686007135256011 c8 Feb 2013 Euler irregular, ECPP 4 - E(2202)/537810555509347782831044328141290207095938 c8 Feb 2013 Euler irregular, ECPP 5 E(2028)/112461539548456847455412 c55 Mar 2011 Euler irregular, ECPP 6 - E(1990)/83382\

085779506247224170162867654734770337416421056719135258 c8 Feb 2013 Euler irregular, ECPP 7 E(1840)/312372820538783689420604121823849344254812 c4 May 2011 Euler irregular, ECPP 8 E(1736)/(55695515 · 75284987831 · 3222089324971117)4498 c4 Jan 2004 Euler irregular, ECPP 9 - E(1466)/1679005322766544173721069526125343992393682 c8 Feb 2013 Euler irregular, ECPP 10 E(1468)/(95 · 217158949445380764696306893 · 597712879321361736404369071)3671 c4 Dec 2003 Euler irregular, ECPP 11 - E(1174)/505505113426970727107950586393323517632829 c8 Feb 2013 Euler irregular, ECPP 12 - E(1142)/6233437695283865492412648122\

953349079446935570718422828539863590139869022408692697 c77 Apr 2015 Euler irregular, ECPP 13 - E(1078)/3618985444390432578 c4 Feb 2002 Euler irregular, ECPP 14 E(1028)/(6415 · 56837916301577)2433 c4 Feb 2002 Euler irregular, ECPP 15 E(1004)/(579851915 · 80533376783)2364 c4 Feb 2002 Euler irregular, ECPP 16 - E(902)/(9756496279 · 314344516832998594237)2069 c4 Jan 2002 Euler irregular, ECPP 17 - E(886)/686892051 c4 Jan 2002 Euler irregular, ECPP 18 E(660)/(19825 · 3318810147166091 · 13270662249997061963929586463495619)1393 c63 Feb 2017 Euler irregular, ECPP 19 E(676)/878618128969410121818976030235415670049335313139\

115048927177891581742982024754755909556741623770151391 c8 Feb 2013 Euler irregular, ECPP 20 - E(638)/(7235862947323 · 11411779188663863 · 526900327479624797)1343 c4 Jan 2002 Euler irregular, ECPP

- The Top 20: Irregular Primes

- Carlitz1954
L. Carlitz, "Note on irregular primes,"Proc. Amer. Math. Soc.,5(1954) 329--331.MR 15,778b- EM1978
R. ErnvallandT. Metsänkylä, "Cyclotomic invariants and e-irregular primes,"Math. Comp.,32(1978) 617--629.MR 80c:12004a- Ernvall1975
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- Ernvall1979
R. Ernvall, "Generalized Bernoulli numbers, generalized irregular primes and class number,"Ann. Univ. Turku. Ser. A,1:178 (1979) 72 pp..MR 80m:12002- Ernvall1983
R. Ernvall, "Generalized irregular primes,"Mathematika,30:1 (1983) 67--73.MR 85g:11022- Gut1950
M. Gut, "Eulersche zahlen und grosser Fermat'sche satz,"Comment. Math. Helv.,24(1950) 73--99.MR 12,243d- Vandiver1940
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

Chris K. Caldwell
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