A search for primes $p$ such that the 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’s Last Theorem. Vandiver proved that if $x^p+y^p=z^p$ has a solution for integers $x,y,z$ with $\gcd (xyz,p)=1$, then it must be that $E_{p-3}\equiv 0 (\bmod p)$. Numerous combinatorial congruences recently obtained by Z.-W. Sun and 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 a congruence which runs significantly faster than other known congruences involving $E_{p-3}$. Applying this, congruence, via a computation in Mathematica 8, shows that there are only three primes less than $10^7$ that satisfy the condition $E_{p-3}\equiv 0 (\bmod p)$ (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 $E_{p-3}\equiv 0 (\bmod p)$.