Lehmer’s Totient Problem on Composite Lehmer Numbers

Determine whether there exists a composite integer n such that φ(n) divides n − 1; equivalently, ascertain the existence or nonexistence of composite Lehmer numbers n with φ(n) | (n − 1).

Background

The paper lists Lehmer’s Problem as an open problem involving Euler’s totient function φ. Lehmer’s Totient Problem asks whether any composite integer n satisfies φ(n) | (n − 1). No such composite is known, and proving existence or nonexistence remains open.

This question is central to understanding extremal divisibility properties of φ and intersects with themes of the paper where precise behavior of φ under iteration and summation is analyzed, particularly for special families like Fermat primes.