Structured Solutions of Prime-Base Binomial Congruences

In this paper, we study the congruence $\left(\genfrac{}{}{0px}{}{qn}{n}\right)\equiv q^n(\mathrm{m}\mathrm{o}\mathrm{d}n)$ for a prime base $q$. Motivated by the OEIS sequence \seqnum{A080469} and the conjectural existence of infinitely many ternary solutions of the form $n=3^{t}p$, we analyze the more general family $n=q^{t}p$, where $p\mathrm{\ne }q$ is prime. Our main result shows that, in this family, the congruence is equivalent to two independent conditions: a congruence modulo $p$ and an inequality in the sum of the digits. This reduces the search for such solutions to factoring an explicit integer and applying a base-$q$ digit-sum filter. We use this criterion to produce new large solutions for $q\in \{2,3,5,7,11\}$. We also prove that square solutions $n=p^2$ are exactly governed by Wieferich primes in base $q$.