A remark on Giuga’s conjecture and Lehmer’s totient problem∗
نویسنده
چکیده
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. ∗MSC Numbers: 11A07, 11N25
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