Euler's Totient Calculator

Euler's totient function phi(n) counts how many whole numbers from 1 up to n share no common factor with n other than 1, and this engine reaches that count without generating a single one of those numbers directly. It works from n's prime factorization: starting with a running total equal to n, every distinct prime that divides n knocks a (1 minus 1 over p) factor out of that total, which is the standard multiplicative identity behind phi. The trial-division search that finds those primes only needs to check candidates up to the square root of n, so even at the input's ceiling of one million it finishes in a small, bounded number of steps rather than testing a million candidates one at a time. Two of the outputs report essentially the same fact from different angles: phi(n) and Coprime Count are numerically identical, because counting the coprimes to n and evaluating phi(n) are the same operation by definition — the second field exists to name the result by what it means, not to add new information beyond the first.

For a prime p, phi(p) is always p minus 1, since every smaller positive integer is automatically coprime to a prime — so the phi(n)/n ratio for a prime gets closer and closer to 1 as the prime grows (phi(7)/7 is 0.857, phi(997)/997 is 0.999) without ever actually reaching it for any finite prime; the ratio only equals exactly 1 at n = 1, a special case the engine handles on its own and which is not itself prime. The ratio shrinks toward zero for numbers built from several small repeated prime factors instead. A fractional entry like 12.7 is silently floored to 12 before anything else runs, and n is clamped between 1 and 1,000,000 regardless of what the field allows you to type.