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 Pascal tiling, that is, a perfect alternation between entries congruent to 0 mod N and the others, if and only if N is prime.

This result yields an alternative proof of the classical congruence LN -1 ≡ 0 mod N for prime N , where LN denotes the N th Lucas number. Within the framework of the 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 mod N . By contrast, for Fibonacci pseudoprimes, the sum remains congruent to 0 mod 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.

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BibTeX

@misc{rousseau2026pascal,
  title = {{Pascal tiling and congruences modulo N in Pascal's triangle}},
  author = {Etienne Rousseau and Guillaume Rousseau},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08343},
  eprint = {2609.08343},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05735606v1},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Pascal tiling and congruences modulo N in Pascal's triangle
AU  - Etienne Rousseau
AU  - Guillaume Rousseau
PY  - 2026
DA  - 2026/09/02
PB  - HAL
DO  - 10.48550/arxiv.2609.08343
UR  - https://hal.science/hal-05735606v1
AB  - 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 Pascal tiling, that is, a perfect alternation between entries congruent to 0 mod N and the others, if and only if N is prime.  This result yields an alternative proof of the classical congruence LN -1 ≡ 0 mod N for prime N , where LN denotes the N th Lucas number. Within the framework of the 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 mod N . By contrast, for Fibonacci pseudoprimes, the sum remains congruent to 0 mod 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.
ER  -

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