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On Congruence Conditions for Primality
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Sherry Gong
Published/Copyright:
July 8, 2010
Abstract
For any k ≥ 0, all prime 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.
Received: 2009-05-22
Revised: 2010-02-28
Accepted: 2010-03-07
Published Online: 2010-07-08
Published in Print: 2010-July
© de Gruyter 2010
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Articles in the same Issue
- Corrigendum and Statement on Priority for: Conjugacy Classes and Class Number
- The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons
- A Multitude of Expressions for the Stirling Numbers of the First Kind
- Infinitely Often Dense Bases for the Integers with a Prescribed Representation Function
- A Note on the Exact Expected Length of the kth Part of a Random Partition
- On Congruence Conditions for Primality
- The Divisibility of an – bn by Powers of n
- Long Arithmetic Progressions in Small Sumsets
- On the Least Common Multiple of Q-Binomial Coefficients
- Flat Cyclotomic Polynomials of Order Four and Higher
- On Vanishing Sums of Distinct Roots of Unity
