Primorial Calculator

About this calculator

The primorial of n, written n#, is the product of all prime numbers less than or equal to n — analogous to the factorial but multiplying only primes. For example 10# = 2 × 3 × 5 × 7 = 210 (the primes up to 10). Primorials grow extremely fast, even faster than factorials of the same argument once several primes are included, since every factor is itself prime rather than a run of consecutive integers.

Primorials show up throughout number theory: Euclid's classic proof that there are infinitely many primes considers a number one more than a primorial (p# + 1), and "primorial primes" (primes of the form p# ± 1) are an active area of prime-hunting research. They're also used to construct numbers with many small prime factors for testing purposes, and in the study of prime gaps and the distribution of primes via the prime-counting function. Students and number-theory hobbyists use this calculator to compute n# for any n and explore these patterns.