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Can $x^3 + 7$ ever be a square integer, for integer $x$ ? If not, how can this be proved ?

I have tested with a quick Python script and have reached $x = $ approx. 14 billion, with no squares yet found.

import gmpy2

x = 1
output_gap = 10**7


while True :
    if x % output_gap == 0 :
        print('x = %s' % f"{x:,}")
    y = x**3 + 7
    if (gmpy2.is_square(y)) :
        print('Square detected at x = %s' % f"{x:,}")
    x += 1
Bill Dubuque
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Ross Ure Anderson
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    See https://math.stackexchange.com/q/514571/42969, and this: https://math.stackexchange.com/a/332360/42969 – Martin R Oct 10 '24 at 15:10
  • $y^2=x^3+7$ Is an Elliptic Curve over Q & It has a Rank Zero so it has only Finitely Many Rational points on a Curve. – Guruprasad Oct 14 '24 at 01:28

0 Answers0