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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Sign up to join this communityConsider the equation:
$p^3-q^2+2=8\cdot q$
Has this equation infinitely many solutions for $p$ and $q$ prime?
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.
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.