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I have a degree-96 even irreducible polynomial $P(x)$ with integer coefficients between 90 and 110 decimal digits long. I would like to find a value of $x$ such that $P(x)$ is an integer square. (Of course the constant term is not a square.)

Is there a technique or algorithm that would let me find such an $x$, or rule out that such an $x$ exists? (I can't find a modular obstruction.) Of course it need not be specific to my case; I list my parameters to give an idea of the scale of my problem.

There is a question asking the same thing about the quadratic case, but this is far harder. In particular, with degree $\ge5$ Galois theory suggests this will be hairy. :-)

Charles
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    This is looking for integer solutions to $y^2= P(x)$ which is the same as finding points on this hyperelliptic curve. By Faltings, there will be at most finitely many such values. There are plenty of "algorithms" (big quotations there) to try bound the number of solutions as well as find them, but given the high genus and large coefficients, they are not likely to work without great modification with large amounts of work - especially computationally. But you might look at some (non)abelian Chabauty methods: https://swc-math.github.io/aws/2020/index.html – mathematics2x2life Jun 16 '22 at 06:34
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    The galois theory has nothing to do with the problem when $P(x)$ is a perfect square. Considering the probable magnitude of the values of a polynomial like that you are interested in it is unlikely that it can produce a perfect square. – Peter Jun 16 '22 at 07:08

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