2 problems

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Apéry congruence conjecture for the second-kind numbers modulo $9$
Second-kind Apéry congruence conjecture. The numbers $A^{(3)}_n$ satisfy the following congruences modulo $9$ :
openapery-numbers congruences ternary-expansions
07
Conjecture on long-period rational points in the Cantor set
Long-period conjecture. For every $K>2/\log(3/2)$ , one has $S_n^{(K)}=S_n$ for all sufficiently large $n$ ; in particular,
opencantor-set diophantine-approximation ternary-expansions
04

Apéry congruence conjecture for the second-kind numbers modulo 999