MathDB

Sun’s Apéry-like polynomial congruence conjecture

For every prime p>3p>3, prove that ∑n=0p−1(2n+1)3vn ⁣(52)2≡6p4−1433p6(modp7)\displaystyle \sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2\equiv 6p^4-\frac{143}{3}p^6\pmod{p^7}, where vn(x)=∑k=0n(nk)(n+kk)xk\displaystyle v_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}x^k.

References

Primary source

arXiv

Additional references

combinatoricsnumber-theoryp-adic-analysis
Posted · edited

Progress summary

Refreshed
Claimed solved

A new paper claims to prove Sun’s specific congruence, but the result has not been independently verified.

The problem concerns a specific Apéry-like polynomial congruence conjectured by Zhi-Hong Sun. A paper by Yu-Tian Li and Zhi-Hong Sun claims to prove it.

September 2026 claimed proof

On September 23, 2026, Li and Sun’s paper reported the congruence modulo p7p^7 for primes p>3p>3, together with weighted identities and related congruences. The retrieved sources contain no independent verification or reported error, so this is a claimed resolution rather than a verified one.

Current status (as of September 2026): The cited congruence is claimed proved modulo p7p^7 for primes p>3p>3, while independent verification and broader Apéry-like conjectures remain open.

Sources

Solutions 0

No solutions have been posted yet.