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Lehmer's totient problem

The totient function φ(n), also called Euler's totient function, is defined as the number of positive integers ≤ n that are relatively prime to (i.e., do not contain any factor in common with) n, where 1 is counted as being relatively prime to all numbers.

Source : Wolfram MathWorld
In other words, it is the number of integers k in the range 1 ≤ kn for which the greatest common divisor gcd(n, k) is equal to 1.

Source : Wikipedia

In 1932, American mathematician D.H Lehmer outlined his 'Totient Problem', which links the totient function to properties of composite numbers.

Lehmer's totient problem asks if there exist any composite numbers n such that φ (n) | (n-1) where φ (n) is the totient function.

Source : Wolfram MathWorld

The problem remains unresolved.

For an example of current research on the problem see : The spanning method and the Lehmer totient problem arXiv: 2003.13055 (math)


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