The Sierpiński Sieve of Nim-varieties and Binomial Coefficients

We consider the variety XY , where instead of multiplication, we take the Nim-product. Its geometry turns out to be the Sierpinski ´ sieve, which is well known to be connected to Pascal’s triangle modulo two. We generalize Nim-sums and -products to what we call q-sums and -products for integers q ≥ 2 (the original case corresponding to q = 2). The Sierpinski ´ sieve also generalizes to so-called q-sieves, and the original relationship extends completely. That is, the geometry of the q-variety XY is a q-sieve. The connection to binomial coefficients, though, only extends in the case where q is prime, and we prove this using theorems of Kummer and Legendre.