6 problems

Zagier's conjecture. Zagier's finite search yielded all the integral solutions to this recurrence.

Two-term supercongruence conjecture. For all primes $p\geq 5$ and all integers $m,r\geq 1$,

Constant-term classification problem. Can one classify all such integer sequences for which

Gauss-congruence conjecture. All sporadic sequences satisfy the Gauss congruences of order $2$ or $3$, depending on the sequence.

Let $s_7$ and $s_{18}$ be the binomial-sum sequences defined above, and let $m,r$ be positive integers. Prime-power supercongruence conjecture. Based on numerical evidence, the fol…

Let … and … with $s_{18}(0)=1$. Cooper's conjecture. For every prime $p\geqslant3$, … and, for every prime $p$, … These congruences are sporadic-sequence analogues of known congrue…