Proof of two conjectural supercongruences involving Catalan-Larcombe-French numbers

In this paper we mainly prove the following two conjectures of Z.-W. Sun [10]: For any odd prime p, we have∑k=0p−1Pk8k≡1+2(−1)(p−1)/2p2Ep−3(modp3),∑k=0p−1Pk16k≡(−1)(p−1)/2−p2Ep−3(modp3), where Pn=∑k=0n(2kk)2(2(n−k)n−k)2(nk) is the n-th Catalan-Larcombe-French number, En are the Euler numbers which are defined byE0=1,En=−∑k=1⌊n/2⌋(n2k)En−2k(n≥1).