On convolutions of Euler numbers

Zhi-Wei Sun

We show that if p is an odd prime then $$\sum _{k=0}^{p-1}{E}_k{E}_{p-1-k}=1(modp)$$ and $$\sum _{k=0}^{p-3}{E}_k{E}_{p-3-k}=(-1{)}^{(p-1)/2}2E_{p-3}(modp),$$ where E_0,E_1,E_2,... are Eul…