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

y^{2} = x^{4} + 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

(*x*^{2})^{2} < *x*^{4} + *x* + 7 < *x*^{4} + 2*x*^{2} + 1 = (*x*^{2} + 1)^{2}

at least if *x* ≥ 0 is sufficiently large. So *y*^{2} would have to lie strictly between two consecutive squares – an impossibility. A similar observation

(*x*^{2} − 1)^{2} = *x*^{4} − 2*x*^{2} + 1 < *x*^{4} + *x* + 7 < (*x*^{2})^{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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