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 $E_{p-3}\equiv 0(\mathrm{mod}p)$. 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 $E_{p-3}\equiv 0(\mathrm{mod}p)$. 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 $E_{p-3}\equiv 0(\mathrm{mod}p)$ (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 $E_{p-3}\equiv 0(\mathrm{mod}p)$. Moreover, we propose a conjecture on the asymptotic estimate of number of primes $p$ in an interval $[x,y]$ such that $E_{p-3}\equiv A(\mathrm{mod}p)$ for some integer $A$ with $|A|\in [K,L]$.

10 pages. We reported that only three primes $p\in \{149,241,2946901\}$ less than $10^7$ satisfy the condition that related Euler numbers $E_{p-3}$ are divisible by $p$. Nevertheless, we conjecture that there are infinitely many such primes

References in corpus (3)

• On congruences related to central binomial coefficients
• On Delannoy numbers and Schröder numbers
• An Extension of a Congruence by Kohnen

Cited by in corpus (2)

• Several generalizations and variations of Chu-Vandermonde identity
• Variations of Kurepa's left factorial hypothesis