Carmichael Function Calculator

About this calculator

The Carmichael function λ(n), sometimes called the reduced totient, gives the smallest positive integer m such that a^m ≡ 1 (mod n) for every integer a that is coprime to n. It plays a similar role to Euler's totient φ(n) — which counts how many integers below n are coprime to n — but λ(n) is often smaller because it finds the true smallest universal exponent rather than just any exponent that happens to work; in fact λ(n) always divides φ(n).

This function matters in modular arithmetic and cryptography: RSA key generation can use λ(n) in place of φ(n) when deriving the private exponent, and because λ(n) is frequently smaller than φ(n), it can produce a smaller valid exponent for the same modulus, which is the basis of the Carmichael-based variant of the RSA key-generation formula. Number theorists also study λ(n) when analyzing the structure of the multiplicative group of integers modulo n and its cyclic subgroup decomposition.

Enter a positive integer n and the calculator returns both λ(n) and φ(n), letting you compare the reduced totient against the standard Euler totient for the same number.