primeproduct

Primeproduct is not a single standard term in mathematics. In practice, it is used informally to refer to several related notions involving products of prime numbers. The most common interpretations are the product of the distinct prime factors of an integer and the primorial, the product of the first n primes.

One interpretation is the product of distinct primes dividing a positive integer n. This is often called the radical of n and is denoted rad(n). For example, if n = 60 = 2^2 · 3 · 5, then rad(60) = 2 · 3 · 5 = 30. The radical is always a squarefree integer and divides n. It is a multiplicative function in the sense that rad(mn) depends on the prime factors of m and n, and rad(mn) = rad(m) · rad(n) / rad(gcd(m,n)).

A more standard use is the primorial, which is the product of the first n primes. It is usually written p_n# or n#, where p_n denotes the nth prime. For example, 5# = 2 · 3 · 5 · 7 · 11 = 2310. Primorials grow very rapidly; by the prime number theorem, log(p_n#) is asymptotic to p_n. Primorials have applications in modular arithmetic, the construction of integers with prescribed prime divisors, and in sieve theory.

A related notion is the product of primes in a specified set or interval, such as the product of all primes in [a, b], denoted ∏_{p ∈ P ∩ [a,b]} p. This is used in analytic and algebraic contexts when restricting attention to primes within a range.

See also: radical function, primorial, prime factorization.