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

Abstract

Let $p>3$ be a prime. Euler numbers $E_{p-3}$ first appeared in H. S. Vandiver's work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers $x,y,z$ with $\gcd(xyz,p)=1$ if $E_{p-3}\equiv 0 (\bmod p)$. Numerous combinatorial congruences recently obtained by Z.-W. Sun and by Z.-H. Sun involve the Euler numbers $E_{p-3}$. This gives a new significance to the primes $p$ for which $E_{p-3}\equiv 0 (\bmod p)$. For the computation of residues of Euler numbers $E_{p-3}$ modulo a prime $p$, we use the congruence which runs significantly faster than other known congruences involving $E_{p-3}$. Applying this congruence, a computation via {\tt Mathematica 8} shows that only three primes less than $10^7$ satisfy the condition $E_{p-3}\equiv 0 (\bmod p)$ (such primes are 149, 241 and 2946901, and they are given as a Sloane's sequence A198245). By using related computational results and statistical considerations similar to those on search for Wieferich and Fibonacci-Wieferich and Wolstenholme primes, we conjecture that there are infinitely many primes $p$ such that $E_{p-3}\equiv 0 (\bmod p)$. Moreover, we propose a conjecture on the asymptotic estimate of number of primes $p$ in an interval $[x,y]$ such that $E_{p-3}\equiv A (\bmod p)$ for some integer $A$ with $|A|\in [K,L]$.