Pascal tiling and congruences modulo N in Pascal's triangle

Abstract

We investigate the properties of matrices obtained from a geometric transformation of the first $N$ rows of Pascal's triangle. For $N > 2$, their congruence properties form a \emph{Pascal tiling}, that is, a perfect alternation between entries congruent to $0 \pmod{N}$ and the others, if and only if $N$ is prime. This result yields an alternative proof of the classical congruence $L_N-1\equiv 0 \pmod{N}$ for prime $N$, where $L_N$ denotes the $N$th Lucas number. Within the framework of the \emph{Pascal tiling theorem}, this congruence can be expressed as a sum of entries lying along a diagonal of one of the matrices under consideration; when $N$ is prime, each of these entries is congruent to $0 \pmod{N}$. By contrast, for Fibonacci pseudoprimes, the sum remains congruent to $0 \pmod{N}$ while at least one of its terms is not. Finally, these results are interpreted in terms of decompositions of binomial coefficients and extended to multinomial coefficients, leading to a study of the associated symmetries. This perspective highlights the case where $N$ is a prime power and clarifies the conditions under which a Pascal tiling arises.