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Let p > 3 be a prime. Euler numbers Ep−3 first appeared in H. S. Vandiver’s work (1940) in connection with the first case of Fermat’s Last Theorem. Vandiver proved that if xp + yp = zp has a solution for integers x, y, z with gcd(xyz, p) = 1, then it must be that Ep−3 ≡ 0 (mod p). Numerous combinatorial congruences recently obtained by Z.-W. Sun and Z.-H. Sun involve the Euler numbers Ep−3. This gives a new significance to the primes p for which Ep−3 ≡ 0 (mod p). For the computation of residues of Euler numbers Ep−3 modulo a prime p, we use a congruence which runs significantly faster than other known congruences involving Ep−3. Applying this, congruence, via a computation in Mathematica 8, shows that there are only three primes less than 107 that satisfy the condition Ep−3 ≡ 0 (mod p) (these primes are 149, 241 and 2946901). By using related computational results and statistical considerations similar to those used for Wieferich, Fibonacci-Wieferich and Wolstenholme primes, we conjecture that there are infinitely many primes p such that Ep−3 ≡ 0 (mod p).

A search for primes 𝑝 such that the Euler number 𝐸_{𝑝-3} is divisible by 𝑝