Does the above Diophantine equation have infinitely many integer solutions ? One such solution is $(x,y) = (4,27)$.
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Wolfram Alpha can only find the solution you gave. – meiji163 Dec 26 '15 at 20:54
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2May I ask why you are interested in this particular equation? – Wojowu Dec 26 '15 at 21:15
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So that you will be familiar with it, in general $y^2=ax^3+bx^2+cx+d$ will only have a *finite* number of integer points. See a theorem by Siegel. – Tito Piezas III Dec 29 '15 at 14:20
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Multiply both sides by $144,$ you get a Mordell curve $U^2 = V^3 - 5616.$
This has only the solutions you knew about
E_-05616: r = 1 t = 1 #III = 1
E(Q) = <(48, 324)>
R = 1.0595282130
2 integral points
1. (48, 324) = 1 * (48, 324)
2. (48, -324) = -(48, 324)
Will Jagy
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