PRIMES GENERATED BY ELLIPTIC CURVES

For a rational elliptic curve in Weierstrass form, Chud- novsky and Chudnovsky considered the likelihood that the denom- inators of the x-coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a ratio- nal point under an isogeny, we use Siegel's Theorem to prove that only nitely many primes will arise. The same question is consid- ered 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.