Congruences modulo $10^3$ for Euler numbers

Abstract

For a non-negative integer $n$, let $E_n$ be the $n$ th Euler number. In this note, for any positive integer $n$, we prove the following congruences:

$$\begin{aligned} {\left\{ \begin{array}{ll} E_{4n} \equiv 380n-375 \pmod {10^3}, \\ E_{4n+2} \equiv -460n+399 \pmod {10^3}. \end{array}\right. } \end{aligned}$$

Our proof is based on induction on $n$ and elementary direct calculations.