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

Author: Romeo Meštrović
Journal: Math. Comp. 83 (2014), 2967-2976
MSC (2010): Primary 11B75, 11A07; Secondary 11B65, 05A10
Published electronically: February 12, 2014
MathSciNet review: 3246818
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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)$.

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Additional Information

Romeo Meštrović
Affiliation: Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro

Keywords: Euler number, $E_{p-3}$, congruence modulo a prime, supercongruence, Fermat quotient
Received by editor(s): December 31, 2012
Received by editor(s) in revised form: January 5, 2013, and February 6, 2013
Published electronically: February 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.