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Auxiliary Polynomials in Number Theory by David Masser

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4

Diophantine equations – Runge’s Method

We make an apparent jump to solve some diophantine equations. One that often turns up in problem-solving sessions is: find all rational integers x, y with

y2 = x4 + x + 7. (4.1)

Someone with too much knowledge might shy away, thinking that it is a question of elliptic curves and singular models into the bargain. Someone more ignorant might notice that

(x2)2 < x4 + x + 7 < x4 + 2x2 + 1 = (x2 + 1)2

at least if x ≥ 0 is sufficiently large. So y2 would have to lie strictly between two consecutive squares – an impossibility. A similar observation

(x2 − 1)2 = x4 − 2x2 + 1 < x4 + x + 7 < (x2)2

rules out x ≤ 0 sufficiently large. On examining small x one finds just

(x, y) = (−7, ±49), (1, ±3), (2, ±5). ...

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