content:mathematics:lehmer_totient_problem

# Wikenigma - an Encyclopedia of Unknowns Wikenigma - an Encyclopedia of the Unknown

# Lehmer's totient problem

The totient functionÏ†(n), also calledEuler's totient function, is defined as the number of positive integers â‰¤nthat 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 integerskin the range 1 â‰¤kâ‰¤nfor 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 numbersnsuch thatÏ†(n) |(n-1)whereÏ†(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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