Goldbach's conjecture
Mirsky's early research concerned number theory. He was particularly interested in the "r"-free numbers, a generalization of the square-free integers consisting of the numbers not divisible by any "r"th power. These numbers are a superset of the prime numbers, and Mirsky proved theorems for them analogous to Vinogradov's theorem, Goldbach's conjecture, and the twin prime conjecture for prime numbers.