Supercongruences motivated by e

In this paper we establish some new supercongruences motivated by the well-known fact limn→∞⁡(1+1/n)n=e. Let p>3 be a prime. We prove that∑k=0p−1(−1/(p+1)k)p+1≡0(modp5) and∑k=0p−1(1/(p−1)k)p−1≡23p4Bp−3(modp5) where B0,B1,B2,… are Bernoulli numbers. We also show that for any a∈Z with p∤a we have∑k=1p−11k(1+ak)k≡−1(modp)and∑k=1p−11k2(1+ak)k≡1+12a(modp).