Alternating sums of Franel and Apéry numbers: proof of two conjectures of Z.-W. Sun

Let f_k = Σ_j C(k,j)³ be the Franel numbers (OEIS A000172) and A_k = Σ_j C(k,j)² C(k+j,j)² the Apéry numbers (OEIS A005259). In his list "Open conjectures on congruences" (2019), Z.-W. Sun conjectured that for every prime p and every positive integer n n⁻² ( Σ_{k 3; V. J. W. Guo and J. Zeng, 2012, for p > 3). This preprint proves both conjectures in full, that is, the whole p-adic towers: (4.4) for every prime p, including p = 2 and p = 3, and (4.1) for every odd prime p. Method. The key object is a q-analogue F_N(q) of the alternating Franel sums, built from the representation f_k = Σ_j C(k,j) C(j,k−j) C(2j,j). For all positive integers m and n (odd or even, and also when 3 divides n) it satisfies the cyclotomic congruences F_{mn}(q) ≡ (n/3) q^((n²−1)/3) F_m(q^(n²)) (mod Φ_n(q)²), a q-analogue of a Lucas–Dwork type congruence. The proof combines the q-Chu–Vandermonde summation (which links F_N(q) to the sums of central q-binomial coefficients), a q-Lucas theorem modulo Φ_n(q)² valid for every n, an identity of R. Tauraso and a theorem of J.-C. Liu and F. Petrov; the latter is extended along the way to all m. Two residue formulas at the p-power roots of unity, one for P(1) and one for P′(1), then turn the congruences for n = p, p², …, p^r into a congruence modulo p^(2r): for p ≥ 5 everything reduces to vanishing Ramanujan sums, for p = 3 the q-analogue is divisible by [3^r]_q², and for p = 2 a joint induction on the values and the derivatives at q = 1 closes the argument. Conjecture (4.1) is then deduced from (4.4) through Strehl's identity A_k = Σ_i C(k,i) C(k+i,i) f_i, a telescoping identity and a transfer lemma for the weights C(N−1,k) C(N+k,k). Consequences. For p = 3 the two theorems also settle two further questions of Sun: n⁻² Σ_{k<n} (−1)^k f_k is a 3-adic integer for every n (conjectured by Sun in 2013), and ν₃( Σ_{k