A search for primes $p$ such that the Euler number $E_{p-3}$ is divisible by $p$
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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
- Email: romeo@ac.me
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 2967-2976
- MSC (2010): Primary 11B75, 11A07; Secondary 11B65, 05A10
- DOI: https://doi.org/10.1090/S0025-5718-2014-02814-7
- MathSciNet review: 3246818