Primes generated by elliptic curves |
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Authors: | Graham Everest Victor Miller Nelson Stephens |
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Institution: | School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom ; Center for Communications Research, Princeton, New Jersey 08540 ; Department of Mathematical and Computer Sciences, Goldsmiths College, London SE14 6NW, United Kingdom |
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Abstract: | For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the -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. |
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