Congruences involving S_k^(2)(16/5) and S_k^(2)(18)

Establish the mod-p^2 congruences and p-adic integrality relations involving S_k^{(2)}(16/5) and S_k^{(2)}(18) stated in Conjecture 4.13.

Conjecture 4.13 is a related variant of Conjecture 4.12, with the quadratic-form parameter 22 replaced by 37.

Conjecture 4.13. Let p > 5 be a prime.

— A family of polynomials and related congruences and series (2505.02767 - Sun, 5 May 2025) in Conjecture 4.13, Section 4

Conjecture 4.16. Let p > 5 be a prime.

— A family of polynomials and related congruences and series (2505.02767 - Sun, 5 May 2025) in Conjecture 4.16, Section 4