On Congruence Conditions for Primality
نویسنده
چکیده
For any k ≥ 0, all primes n satisfy the congruence nσk(n) ≡ 2 mod φ(n). We show that this congruence in fact characterizes the primes, in the sense that it is satisfied by only finitely many composite n. This characterization generalizes the results of Lescot and Subbarao for the cases k = 0 and k = 1. For 0 ≤ k ≤ 14, we enumerate the composite n satisfying the congruence. We also prove that any composite n which satisfies the congruence for some k satisfies it for infinitely many k.
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