Euler Irregular primes

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[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 < 2n < 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 is Euler-irregular (E-irregular) if and only if it divides an Euler number E(2n) with 0 < 2n < p-1.

Euler numbers are obtained from Taylor coefficients of

1 / cosh x = sum_{k > 0} 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

(n) = sum_{k > 0} 1/kⁿ

which evaluates to

(2n) = - (-4²)ⁿ*B(2n)/(2(2n)!)

at the even positive integers. E-irregularity entails the unique character modulo 4, namely \chi(k)=sin(k/2), with the corresponding Dirichlet series

\beta(n) = sum_{k > 0} sin(k/2)/kⁿ

evaluating to

\beta(2n+1) = (-²/4)ⁿ E(2n)/(4(2n)!)

at the odd positive integers, with Gregory's formula

/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...

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) = e^-1/2(1/2)^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 < 2n < p-1 < 10,000 and E(2n)=0 (mod 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 = 0	k = 1	k = 2	k = 3	k > 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)
-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)

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.