Structured Solutions of Prime-Base Binomial Congruences

In this paper, we study the congruence $\binom{qn}{n} \equiv q^n \pmod n$ 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 $p\neq q$ 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 $q$.