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arXiv · 1108.3830

Elliptic Carmichael Numbers and Elliptic Korselt Criteria

Joseph H. Silverman
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Abstract

Let E/Q be an elliptic curve, let L(E,s)=\sum a_n/n^s be the L-series of E/Q, and let P be a point in E(Q). An integer n > 2 having at least two distinct prime factors will be be called an elliptic pseudoprime for (E,P) if E has good reduction at all primes dividing n and (n+1-a_n)P = 0 (mod n). Then n is an elliptic Carmichael number for E if n is an elliptic pseudoprime for every P in E(Z/nZ). In this note we describe two elliptic analogues of Korselt's criterion for Carmichael numbers, and we analyze elliptic Carmichael numbers of the form pq.

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Joseph H. Silverman. 2011-08-18. Elliptic Carmichael Numbers and Elliptic Korselt Criteria. https://arxiv.org/abs/1108.3830

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