Supercongruences motivated by e

TL;DR

This paper proves new supercongruences related to the exponential limit e, involving binomial sums and Bernoulli numbers, extending the understanding of congruences modulo prime powers.

Contribution

It establishes novel supercongruences involving binomial coefficients, Bernoulli numbers, and exponential sums, motivated by the limit definition of e.

Findings

Proved that certain binomial sum is congruent to 0 modulo p^5.

Derived a congruence involving Bernoulli numbers and binomial sums modulo p^5.

Established congruences for exponential sums involving reciprocals modulo p.

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 $B_0,B_1,B_2,\ldots$ are Bernoulli numbers. We also show that for any $a\in\mathbb Z$ with $p\nmid a$ we have $$\sum_{k=1}^{p-1}\frac1k\left(1+\frac ak\right)^k\equiv -1\pmod{p}\ \ \ \mbox{and}\ \ \ \sum_{k=1}^{p-1}\frac1{k^2}\left(1+\frac ak\right)^k\equiv 1+\frac 1{2a}\pmod{p}.$$