Divisibility - with visibility
If one reduces all the entries in the Pascal triangle of binomials modulo 2 or more generally modulo a prime number p the resulting triangle of residuals shows interesting patterns which can be visualized by the use of computer outprints. Exploiting these regularities leads to well comprehensible proofs of deeper relations for the binomials like a theorem by Lucas. This approach can then be transferred and generalized to other arrays of combinatorial numbers which obey recurrence relations of a structure similar to that for the binomials. This is demonstrated here for the Gaussian coefficients (number of subspaces in finite vector spaces) and the Stirling numbers of second kind. Aesthetically attractive computer outprints are shown further for the case of a prime power as a modulus. There are a number of open problems in these cases as there are for higher dimensional arrays (like for the multinomials).
scientific article; zbMATH DE number 4049650