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 A173200 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0,1,2, ... (solutions x are given by A053755). 2
 0, 11, 70, 225, 524, 1015, 1746, 2765, 4120, 5859, 8030, 10681, 13860, 17615, 21994, 27045, 32816, 39355, 46710, 54929, 64060, 74151, 85250, 97405, 110664, 125075, 140686, 157545, 175700, 195199, 216090, 238421, 262240, 287595, 314534, 343105 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For many values of k for the equation y^2 = x^3 + k, all the solutions are known. For example, we have solutions for k=-2: (x,y) = (3,-5) and (3,5). A complete resolution for all integers k is unknown. Theorem: Let k be < -1, free of square factors, with k == 2 or 3 (mod 4). Suppose that the number of classes h(Q(sqrt(k))) is not divisible by 3. Then the equation y^2 = x^3 + k admits integer solutions if and only if k = 1 - 3a^2 or 1 - 3a^2 where a is integer. In this case, the solutions are x = a^2 - k, y = a(a^2 + 3k) or -a(a^2 + 3k) (the first reference gives the proof of this theorem). With k = -1 - 3a^2, we obtain the solutions x = 4a^2 + 1, y = a(8a^2 + 3) or -a(8a^2 + 3). For the case k = 1 - 3a^2, we obtain the solution x = 4a^2 - 1 given by the sequence A000466. REFERENCES T. Apostol, Introduction to Analytic Number Theory, Springer, 1976 D. Duverney, Théorie des nombres (2e edition), Dunod, 2007, p.151 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall & D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory 4 (1972), 107-117. John J. O'Connor and Edmund F. Robertson, Louis Joel Mordell Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k (solutions for 0

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Last modified July 5 01:22 EDT 2020. Contains 335457 sequences. (Running on oeis4.)