Lambda function

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In number theory, the Lambda function is a function on positive integers which gives the exponent of the multiplicative group modulo that integer.

The value of λ on a prime power is:

  • \lambda(2) = 1;~ \lambda(4) = 2;~ \lambda(2^n) = 2^{n-2} \mbox{ for } n \ge 2; \,
  • \lambda(p^n) = p^{n-1}(p-1) \mbox{ for } n \ge 1 \, if p\, is an odd prime.

The value of λ on a general integer n with prime factorisation

n = \prod_i p_i^{a_i} \,

is then

\lambda(n) = \mathop{\mbox{lcm}}_i \{ \lambda(p_i^{a_i}) \} .\,

The value of λ(n) always divides the value of Euler's totient function φ(n): they are equal if and only if n has a primitive root.

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