6

The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn't have any answer and its proof can be obtained by using quadratic reciprocity law.

Do you know answers of this equation for two or three different values of $k$? In addition, do you know any reference about that?

BenyaminH
  • 107
  • 3
    This question could sure use a little effort on the format aspect (sentence structure, LaTex, etc). – barak manos Jan 16 '15 at 20:34
  • 3
    I tried to edit your question, but some passages were too unclear to attempt an edit. Could you clarify the question? – rubik Jan 16 '15 at 20:37
  • 2
    The abrupt mention of $p$ needs clarification. Is $p$ supposed to be $k$, or a prime factor of $k$? – hardmath Jan 16 '15 at 20:39
  • 1
    I wish more people knew what they were missing out on by not knowing how to TeX! – Daniel W. Farlow Jan 16 '15 at 20:41
  • 2
    "As you know": that is not a good way to start, as many who read the question will not know. I think "count $m$" is supposed to be "divides $m$." – KCd Jan 16 '15 at 20:47
  • 1
    http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf See this paper. These are Mordell's equations and can also sometimes be solved using only elementary unique factorization in $\mathbb Z$ or sometimes by using the UFDs $\mathbb Z\left[\sqrt{-1}\right]$, $\mathbb Z\left[\sqrt{-2}\right]$. – user263326 Aug 04 '16 at 19:03

1 Answers1

6

This is a famous class of elliptic curves, called Mordell's equation, or sometimes Mordell-Bachet equation. See also here, or here for some discussions on MSE. For a specific example with $k=2000000$ see also here. A further reference is this article by Keith Conrad.

Community
  • 1
Dietrich Burde
  • 130,978