Proving Only 2 Positive Integer Solutions for x3 = y2 – 15

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A problem asks to find all possible pairs (x, y) of positive integers that satisfy the equation:

x3 = y2 – 15

There are 2 pairs (so far) that satisfy the equation:

x = 1, y = 4
x = 109, y = 1138

It's possible that these 2 points are the only two positive integer solutions.

Siegel's theorem states that an elliptic curve can have only a finite number of points with integer coordinates.

Could there be other points for that curve? If not, how to prove that these 2 points are the only solutions?
 
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Siegel's theorem does not help you here. You need to find a method that let's you set some kind of quantifiable limit on the ratio of something to something. Wikipedia references something called "Baker's method", which could be worth investigating. Then you'd prove that the points you found are the only solutions, by proving that there are no solutions above X, and by checking all points up to X manually.
 

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