====== 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 : [[https://mathworld.wolfram.com/TotientFunction.html|Wolfram MathWorld]] \\

>In other words, it is the number of integers //k// in the range 1 ≤ //k// ≤ //n// for which the greatest common divisor gcd(//n//, //k//) is equal to 1.\\ \\ Source : [[https://en.wikipedia.org/wiki/Euler%27s_totient_function|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 : [[https://mathworld.wolfram.com/LehmersTotientProblem.html|Wolfram MathWorld]]

The problem remains unresolved.

For an example of current research on the problem see : [[https://arxiv.org/abs/2003.13055|The spanning method and the Lehmer totient problem]] arXiv: 2003.13055 (math)