Supercongruences motivated by e

Abstract

In this paper we establish some new supercongruences motivated by the well-known fact ${\lim }_{n\to \infty }(1+1/n{)}^n=e$. Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{-1/(p+1)}k^{p+1}\equiv 0\ \pmod{p^5}\ \ \ \mbox{and}\ \ \ \sum_{k=0}^{p-1}\binom{1/(p-1)}k^{p-1}\equiv \frac{2}{3}p^4B_{p-3}\ \pmod{p^5},$$ where $B_0,B_1,B_2,\dots$ are Bernoulli numbers. We also show that for any $a\in \mathbb{Z}$ with $p\nmid a$ we have $$\sum_{k=1}^{p-1}\frac1k\left(1+\frac ak\right)^k\equiv -1\pmod{p}\ \ \ \mbox{and}\ \ \ \sum_{k=1}^{p-1}\frac1{k^2}\left(1+\frac ak\right)^k\equiv 1+\frac 1{2a}\pmod{p}.$$

16 pages

References in corpus (1)

• Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$