Consider the equation:
$p^3-q^2+2=8\cdot q$
Has this equation infinitely many solutions for $p$ and $q$ prime?
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9$\begingroup$ No, every Mordell curve has only finitely many integral points. $\endgroup$– Bart MichelsCommented Mar 25, 2020 at 20:25
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$\begingroup$ Actually, there are no prime solutions at all. See my response. $\endgroup$– GH from MOCommented Mar 26, 2020 at 6:33
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$\begingroup$ Every elliptic curve over a number field has only finitely many integral points. This was proven by Carl Ludwig Siegel. See chapter IX of Silverman's book "The Arithmetic of Elliptic Curves". $\endgroup$– The Thin WhistlerCommented Mar 26, 2020 at 8:37
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4$\begingroup$ I'm voting to close this question because (a) there was no motivation provided or context for the question (b) the user did not engage with the comments and answers (c) the user has not been active on this account, and seems to instead be creating new burner accounts every few days $\endgroup$– Yemon ChoiCommented Apr 4, 2020 at 5:03
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2 Answers
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According to SAGE, the integral solutions are $(7,-23)$ and $(7,15)$:
sage: EllipticCurve([0,0,8,0,2]).integral_points(both_signs=True)
[(7 : -23 : 1), (7 : 15 : 1)]
Hence there are no (positive) prime solutions.
answered Mar 26, 2020 at 6:15
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Actually there is no solution at all.
[8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389] [18, 31, 63, 103, 207, 271, 423, 511, 711, 1071]
For the first 10 primes, evaluated at ^3 and ^2 + 8* - 2 respectively. After that the ^3 goes way much faster than the other so no solution at all.
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2$\begingroup$ You seem to be assuming that $p$ and $q$ are equal. $\endgroup$ Commented Mar 26, 2020 at 6:01