primesquare

Primesquare is the term used for the square of a prime number. If p is prime, then primesquare refers to the number p^2. Examples include 4 (2^2), 9 (3^2), 25 (5^2), 49 (7^2), and 121 (11^2).

As a class of numbers, primesquares are perfect squares that are not prime themselves. Their prime factorization consists of a single prime raised to the second power, p^2, so they are always composite for p > 1. A characteristic property is that they have exactly three positive divisors: 1, p, and p^2. Consequently, primesquares are not squarefree, since their prime exponent is 2.

In terms of distribution, the count of primesquares up to a value x equals the number of primes up to sqrt(x). Therefore, the asymptotic density of primesquares is governed by the distribution of primes and is roughly sqrt(x) / log sqrt(x) as x grows large.

Identification and computation are straightforward: to test whether a number n is a primesquare, check if n is a perfect square; if so, let r be its integer square root. If r is prime, then n is a primesquare. This makes primesquares easy to recognize in practice and useful in discussions about numbers with exactly three divisors or in studies of prime powers.

Related concepts include the square of primes in general, squareful numbers, and the broader study of prime powers. The sequence of primesquares begins 4, 9, 25, 49, 121, 169, 289, and continues with p^2 for successive primes p.