primesquares

Primesquares, also called prime squares, denote the integers of the form p^2 where p is a prime number. In other words, they are the squares of primes. Since every prime p ≥ 2, p^2 ≥ 4, and none of these numbers is prime; every prime square is composite.

Each prime square has the prime factorization p^2 and exactly three positive divisors: 1, p, and p^2. For example, 4 = 2^2, 9 = 3^2, 25 = 5^2, 49 = 7^2, and 121 = 11^2.

The set of prime squares is infinite because there are infinitely many primes. The number of prime squares up to x equals the number of primes ≤ sqrt(x), so it grows like pi(sqrt(x)) ~ sqrt(x) / log sqrt(x). Consequently, prime squares become sparser among integers as numbers grow larger.

Terminology: prime square, primesquare, and prime-squared are used interchangeably. They should not be confused with squares of composite numbers or with primes themselves. They can appear in elementary number theory problems and in discussions of divisor functions, since a prime square has three divisors, unlike most other squares.