A REMARK ON GIUGA’S CONJECTURE AND LEHMER’S TOTIENT PROBLEM

Abstract

Giuga has conjectured that if the sum of the (n−1)-st powers of the residues modulo n is −1(mod n), then n is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if φ(n) divides n−1, with φ 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 φ(n)|n−1.

Locations

• Albanian Journal of Mathematics - View - PDF