Radically Weakening the Lehmer and Carmichael Conditions

Abstract

Lehmer's totient problem asks if there exist composite integers n satisfying the condition ϕ(n)|(n − 1), (where ϕ is the Eulerphi function) while Carmichael numbers satisfy the weaker condition λ(n)|(n−1) (where λ is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of ϕ(n) also divides n − 1. (So rad(ϕ(n))|(n − 1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound which is heuristically tight.

Carmichael numbers and the sieve

“…The conjecture was proved unconditionally by Matomäki [20] in the special case that a is a quadratic residue modulo b, and using an extension of her methods Wright [23] established the conjecture in full generality. The techniques introduced in [1] have led to many other investigations into the arithmetic properties of Carmichael numbers; see [2][3][4][5]7,8,10,[14][15][16][17][18][19]21,24] and the references therein.…”