On Composite Numbers of the Form $p_{1}p_{2} \ldots p_{k} – 1$

conjectureselementary-number-theoryfactoringnumber theoryprime numbers

This question is related to D. H. Lehmer's 1932 conjecture on Euler's totient function: Are there any composite $n$ for which $\phi(n)$ divides $n-1$?

See, for example:

On Lehmer's Totient Conjecture

I would like to ask what is known regarding the factors of $p_{1}p_{2} \ldots p_{k} – 1$, where the $p_{i}$ are distinct odd prime numbers?

I was not able to find much in some number theory books I consulted.

Thank you.

Best Answer

Here are some additional things related Lehmer's totient conjecture that is known.

Definition: If $n$ is composite then $\phi(n)<n−1$, hence there is at least one divisor $d$ of $n−1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trivially, if $n$ is prime then it has no totient divisor and if $n−1$ is prime then $n$ has exactly 1 totient divisor.

  1. There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$.
  2. Odd numbers prefer not to have a prime number of totient divisors.
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