Hardy-Ramanujan Theorem

The Hardy-Ramanujan Theorem states that, as numbers become very large, the proportion of numbers that are perfect squares or perfect cubes becomes extremely small. In other words, among large numbers, most are not perfect powers. For example, perfect squares (like 4, 9, 16) or perfect cubes (like 8, 27, 64) are relatively rare compared to the total set of large numbers. As numbers grow larger infinitely, the chance of randomly picking a perfect power approaches zero, highlighting how “rare” these special numbers become in the vast landscape of natural numbers.