2018/08/11 by Seoyoung C. Kim, Kim, Seoyoung, Alexandra Walsh +1
Computer Science · Mathematics · #11B37 #11B39 #11G05 #11Y11 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1808.03846
openalex publication_date 2018/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a pair (E,P) of an elliptic curve E/ℚ and a nontorsion point P∈ E(ℚ), the sequence of elliptic Fermat numbers is defined by taking quotients of terms in the corresponding elliptic divisibility sequence (Dn)n∈ℕ with index powers of two, i.e. D1, D2/D1, D4/D2, etc. Elliptic Fermat numbers share many properties with the classical Fermat numbers, Fk=22k+1. In the present paper, we show that for magnified elliptic Fermat sequences, only finitely many terms are prime. We also define generalized elliptic Fermat numbers by taking quotients of terms in elliptic divisibility sequences that correspond to powers of any integer m, and show that many of the classical Fermat properties, including coprimality, order universality and compositeness, still hold.