arXiv 1212.36022012-12-14math.NT

寻找素数$p$使得欧拉数$E_{p-3}$可被$p$整除

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

Romeo Mestrovic

备注：: 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
发表信息：: Mathematics of Computation 83, No. 290 (2014), 2967-2976

摘要：

让$p>3$是一个素数。欧拉数$E_{p-3}$最早出现在H. S. Vandiver的工作（1940年）与费马大定理的第一个案例。Vandiver证明了$x^p+y^p=z^p$对整数$x,y,z$和$\gcd(xyz,p)=1$无解，如果$E_{p-3}\equiv 0 (\bmod p)$ .最近由Z.- W.太阳和Z。H. Sun涉及欧拉数$E_{p-3}$。这给了素数$p$一个新的意义，其中$E_{p-3}\equiv 0 (\bmod p)$。为了计算欧拉数$E_{p-3}$模一个素数$p$的余数，我们使用的同余式运行速度明显快于其他已知的涉及$E_{p-3}$的同余式。应用这个同余，通过{ \tt Mathematica 8}的计算表明，只有三个小于$10^7$的素数满足条件$E_{p-3}\equiv 0 (\bmod p)$（这些素数是149，241和2946901，它们被给出为斯隆序列A198245）。利用与寻找Wieferich素数、Fibonacci-Wieferich素数和Wolstenholme素数类似的计算结果和统计考虑，我们猜想存在无穷多个素数$p$使得$E_{p-3}\equiv 0 (\bmod p)$ .此外，我们还提出了一个猜想，关于素数个数$p$在区间PDMATHAMV中的渐近估计，使得对于某个整数$A$，$E_{p-3}\equiv A (\bmod p)$满足$|A|\in [K,L]$。

英文摘要：

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

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