### A remark on Giuga's conjecture and Lehmer's totient problem

*William D. Banks, C. Wesley Nevans, Carl Pomerance*

#### Abstract

Giuga has conjectured that if the sum of the $(n-1)$-st powers of the residues modulo $n$ is $-1\pmod n$, then $n$ is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if $\varphi(n)$ divides $n-1$, with $\varphi$ being Euler's function, must it be true that $n$ is 1 or prime. No examples are known, but a composite number with this property must be a Carmichael number. We show that there are infinitely many Carmichael numbers $n$ that are not counterexamples to Giuga's conjecture and also do not satisfy $\varphi(n)\mid n-1$.

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