productofprimes

In number theory, the product of primes usually refers to an integer obtained by multiplying prime numbers together. By the fundamental theorem of arithmetic, every integer greater than 1 can be uniquely expressed as a product of primes raised to nonnegative powers.

If n = p1^{a1} p2^{a2} ... pk^{ak} is its prime factorization, then n is the product of primes with multiplicities. If one takes each distinct prime that divides n exactly once, the result is the radical rad(n) = p1 p2 ... pk. The primorial p# denotes the product of all primes up to p.

Examples: 60 = 2^2 · 3 · 5 is a product of primes; its radical is 2 · 3 · 5 = 30. The primorial 7# equals 210.

Properties: The factorization is unique; each n > 1 has finitely many prime factors. 1 is not usually regarded as a product of primes, though an empty product is sometimes defined as 1.

Asymptotics: The product of all primes not exceeding x is the primorial x#, with log x# = θ(x) = ∑_{p≤x} log p, and θ(x) ~ x by the prime number theorem. Consequently, x# grows roughly like e^{x} as x becomes large.

Applications: The concept underpins prime factorization, gcd and lcm calculations, cryptography, and various theoretical results about the distribution of primes and the structure of integers.