Stochastic Domination of Prime Powers of a Uniform Random Integer by Geometric Distributions - Squillace, Joseph; George, Carissa (The PUMP Journal of Undergraduate Research, 2025)
For each natural number n ≥ 2 and for each prime p ≤ n, we provide three proofs of the fact that the power, Cp(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, Zp, of parameter 1/p. In one of these proofs, we construct a coupling of Zp and Cp(n) such that with probability one we have both Zp-1 ≤ Cp(n) ≤ Zp whenever Zp ≤ ⌊logp n⌋ and Cp(n) = ⌊logp n⌋ whenever Zp > ⌊logp n⌋; then we will show that any coupling of Zp and Cp (n) satisfying this constraint is a maximal coupling of Zp and Cp(n) if and only if n = pk for some positive integer k. We will also show how our couplings of the variables Zp and Cp(n) correspond to rigid bracings of an ∞ × ⌊logp n⌋ rectangular grid if and only if n is not divisible by p.