Inputs
A positive integer, arbitrarily large.
Factorization difficulty depends on the size of the factors, not just the size of the input. The calculator may return a partial factorization for difficult inputs. Computation runs locally in your browser.
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Result
Prime factorization
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Factorization incomplete
φ(n) — Euler Totient
Related:
How it works
For a prime power, the integers from 1 to p^a not coprime to p^a are exactly the multiples of p, of which there are p^(a−1). So
φ(p^a) = p^a − p^(a−1) = p^(a−1)(p − 1).
φ is multiplicative: φ(mn) = φ(m)φ(n) when gcd(m, n) = 1. Hence for n = p₁^a₁ ⋯ p_r^a_r,
φ(n) = ∏ p_i^(a_i−1)(p_i − 1) = n ∏(1 − 1/p_i).
The calculator uses only integer arithmetic. By convention φ(1) = 1.