%0 Journal Article
%T A remark on Giuga's conjecture and Lehmer's totient problem
%A William D. Banks
%A C. Wesley Nevans
%A Carl Pomerance
%J Albanian Journal of Mathematics
%D 2009
%I AulonaPress
%X Giuga has conjectured that if the sum of the $(n-1)$-st powers of the residues modulo $n$ is $-1pmod n$, 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 $varphi(n)mid n-1$.
%U http://journals.aulonapress.com/index.php/ajm/article/view/132/113