factorsin

Factorsin is a term sometimes used in algebra to denote the collection of factors that appear in the factorization of a number or a polynomial. It refers to the irreducible factors along with their multiplicities. For integers, if n has prime factorization n = p1^e1 p2^e2 ... pk^ek, then factorsin(n) can be represented as the multiset {p1^e1, p2^e2, ..., pk^ek} or, equivalently, as the list of prime powers with their multiplicities. For example, factorsin(360) = {2^3, 3^2, 5}. The product of all elements in factorsin(n) equals n, and the exponents reflect multiplicity.

For polynomials over a unique factorization domain, a similar idea applies. If f(x) factors as f(x) = ∏ q_j(x)^{t_j} with irreducible q_j, then factorsin(f) = {q_j^{t_j}}. This provides a compact summary of the factorization structure without listing every term.

Relation to related concepts: the radical of n, rad(n) = ∏ p_i, is the product of the distinct primes in the factorsin(n); the exponents in factorsin carry multiplicity. In terms of divisibility, if a divides b, then the factorsin of a is contained in the factorsin of b, respecting multiplicities.

Computationally, obtaining factorsin requires factorization, via methods such as trial division or Pollard’s rho for integers, and standard algorithms for polynomial factorization over fields. Factoring plays a central role in number theory and cryptography, as well as in symbolic computation.