Supercongruence

A supercongruence is a congruence modulo a high power of a prime number, typically strengthening a corresponding congruence modulo that prime number (Zudilin 2009). The term often applies to truncated hypergeometric series and sums of binomial coefficients. For example, Wolstenholme's theorem implies

(1)

where is a prime number (Guy 1994).

Sun's three conjectures 3.5, 3.6, and 3.7 (Sun 2022) concern the sequences

(2)

and

(3)

where is the th Franel number. For a prime number and a denominator not divisible by , write

(4)

and

(5)

The rational numbers in these sums are interpreted modulo by inverting their denominators. Put for odd . In the following formulas, and are integers.

Babanskyy (2026a) reported a proof of Sun's conjecture 3.5, which asserts for every prime number that

$epsilon_pS_p(54)={4x^2-2p (mod p^2) if p=x^2+3y^2 and p=1 (mod 3); 0 (mod p^2) if p=2 (mod 3).$

(6)

Babanskyy (2026b) reported a proof of Sun's conjecture 3.6, which asserts for every odd prime number that

$T_p(-32)=T_p(64)={4x^2-2p (mod p^2) if p=x^2+2y^2 and p=1,3 (mod 8); 0 (mod p^2) if p=5,7 (mod 8).$

(7)

Babanskyy (2026c) reported a proof of Sun's conjecture 3.7, which asserts for every prime number other than 2 and 5 that

$epsilon_pS_p(20)=epsilon_pT_p(-16)={4x^2-2p (mod p^2) if p=x^2+5y^2 and p=1,9 (mod 20); 2x^2-2p (mod p^2) if 2p=x^2+5y^2 and p=3,7 (mod 20); 0 (mod p^2) if p=11,13,17,19 (mod 20).$

(8)

The author reports assistance from OpenAI Codex and GPT-6.1 Sol in proof exploration and checking. As of Oct. 5, 2026, VibeMathed had independently checked the formulas for eligible primes below 260. These finite checks did not establish the general arguments, and independent specialist review and a complete priority check had not been reported. No Lean formalizations were supplied (VibeMathed 2026a, VibeMathed 2026b, VibeMathed 2026c).