A Dwork-type tower for the divergent Ramanujan-type sums Σ(3k+1)C(2k,k)³/16^k: proof of Guo's conjecture for p ≥ 5 and its extension to all m

Let S(N) = Σ_{k 3. Result. We prove that for every prime p ≥ 5 and all integers m, r ≥ 1,S(m p^r) ≡ p S(m p^(r−1)) (mod p^(4r)),and more precisely v_p(S(pN) − p S(N)) ≥ 4 + 4 v_p(N) + 2 v_p(C(2N,N)) for every N ≥ 1. The case m = 1 is Guo's conjecture for p ≥ 5. The exponent 4r is attained (for instance v_5(S(5^r) − 5 S(5^(r−1))) = 4r for r ≤ 6), and p = 3 must be excluded. Method. The starting point is the identity S(n) = 4n² c(n)² Σ_{k