{"id":"integer-totient","version":"1.0.0","description":"Compute Euler's totient function: the count of integers in [1, value] coprime to value.","supported_operations":["integer totient","euler's totient function","euler phi function","count coprime integers"],"unsupported_operations":["totient of 0 or negative integers","carmichael function","totient summatory function"],"semantics":["value is a canonical positive integer string from 1 to 1000000000000 (10^12) inclusive.","0 and negative values are rejected: the totient is only defined here for positive integers.","value 1 returns 1 by convention (the empty product / one coprime residue).","Computed exactly from the prime factorization as value * product over distinct primes p dividing value of (1 - 1/p), using integer arithmetic only (no floating point).","Derivation uses exact trial division up to sqrt(value), bounded by the 10^12 input cap for predictable running time."],"limits":{"max_value":"1000000000000","max_digits":13},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["value"],"properties":{"value":{"type":"string","pattern":"^[1-9][0-9]*$","maxLength":13}}},"output_schema":{"type":"object","additionalProperties":false,"required":["value"],"properties":{"value":{"type":"string"}}},"examples":[{"input":{"value":"36"},"output":{"value":"12"}}],"execute_url":"/v1/tools/integer-totient/versions/1.0.0/execute"}