An extension of Stern's congruence

TL;DR

This paper extends Stern's congruence by determining the Euler numbers' differences modulo a high power of two, providing new insights into their divisibility properties.

Contribution

It generalizes Stern's congruence to a broader class of Euler numbers with explicit modular formulas for their differences.

Findings

Derived explicit formulas for $E_{2^mk+b}-E_b$ modulo $2^{m+7}$

Extended the understanding of Euler numbers' divisibility properties

Provided a new framework for analyzing Euler numbers in modular arithmetic

Abstract

Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.