phi64320

phi64320 denotes Euler's totient function evaluated at 64320, i.e., phi(64320). In number theory, phi(n) counts the integers from 1 to n that are relatively prime to n. Thus phi64320 equals the number of integers in that range that share no common factor with 64320 besides 1.

The value is determined from the prime factorization of 64320. It factors as 64320 = 2^6 × 3 × 5 × 67. Euler's totient function for such a product is phi(n) = n × (1 − 1/2) × (1 − 1/3) × (1 − 1/5) × (1 − 1/67). Substituting the factors gives phi(64320) = 64320 × (1/2) × (2/3) × (4/5) × (66/67) = 16,896. Therefore, phi64320 = 16,896.

Properties of phi64320 include that it is the order of the multiplicative group of integers modulo 64320, meaning there are 16,896 integers less than or equal to 64320 that are coprime to 64320. This value also satisfies Euler's theorem: for any integer a coprime to 64320, a^phi64320 ≡ 1 mod 64320.

Applications of totients, including phi64320, appear in modular arithmetic, cryptography, and the study of multiplicative structures modulo n. While 64320 is not a typical modulus used in standard RSA deployments, phi64320 serves as a concrete example of how totients are computed and applied.

See also: Euler's totient function, totatives, Carmichael function. History: Euler introduced the totient function in the 18th century as part of work on number-theoretic congruences.