The Sierpiński sieve of Nim-varieties and binomial coefficients
Aviezri S. Fraenkel, Alex Kontorovich · 2012
We consider the variety XY, where instead of multiplication, we take the Nim-product. Its geometry turns out to be the Sierpiński 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 Sierpiński 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.