Stochastic Domination of Prime Powers of a Uniform Random Integer by Geometric Distributions

Abstract

For each natural number n ≥ 2 and for each prime p ≤ n, we provide three proofs of the fact that the power, C_p(n), of the prime p in the prime factorization of a uniformly chosen random integer from 1 to n is stochastically dominated by a nonnegative geometric random variable, Z_p, of parameter 1/p. In one of these proofs, we construct a coupling of Z_p and C_p(n) such that with probability one we have both Z_p-1 ≤ C_p(n) ≤ Z_p whenever  Z_p ≤ ⌊log_p n⌋ and C_p(n) = ⌊log_p n⌋ whenever Z_p > ⌊log_p n⌋; then we will show that any coupling of Z_p and C_p (n) satisfying this constraint is a maximal coupling of Z_p and C_p(n) if and only if n = p^k for some positive integer k. We will also  show how our couplings of the variables Z_p and C_p(n) correspond to rigid bracings of an ∞ × ⌊log_p n⌋ rectangular grid if and only if n is not divisible by p