English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 LastTheorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers$x,y,z$ with $\gcd(xyz,p)=1$ if $E_{p-3}\equiv 0 (\bmod p)$. Numerouscombinatorial congruences recently obtained by Z.-W. Sun and by Z.-H. Suninvolve the Euler numbers $E_{p-3}$. This gives a new significance to theprimes $p$ 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 knowncongruences involving $E_{p-3}$. Applying this congruence, a computation via{\tt Mathematica 8} shows that only three primes less than $10^7$ satisfy thecondition $E_{p-3}\equiv 0 (\bmod p)$ (such primes are 149, 241 and 2946901,and they are given as a Sloane's sequence A198245). By using relatedcomputational results and statistical considerations similar to those on searchfor Wieferich and Fibonacci-Wieferich and Wolstenholme primes, we conjecturethat there are infinitely many primes $p$ such that $E_{p-3}\equiv 0 (\bmodp)$. Moreover, we propose a conjecture on the asymptotic estimate of number ofprimes $p$ in an interval $[x,y]$ such that $E_{p-3}\equiv A (\bmod p)$ forsome integer $A$ with $|A|\in [K,L]$.

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