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Primes generated by elliptic curves

2003; American Mathematical Society; Volume: 132; Issue: 4 Linguagem: Inglês

10.1090/s0002-9939-03-07311-8

1088-6826

Graham Everest, Victor S. Miller, N. M. Stephens,

Advanced Algebra and Geometry

For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the x x -coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel’s Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan’s famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.