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In a book "Which Way did the Bicycle Go" there is the following problem:

Find a quadratic polynomial with integer coefficients $p(x)=ax^2+bx+c$ such that $p(1),p(2),p(3),$ and $p(4)$ are perfect squares but $p(5)$ is not.

Does anyone knows some generalizations of this? For example, are the following problems open:

  1. Find all quadratic polynomial with integer coefficients $p(x)=ax^2+bx+c$ such that $p(1),p(2),p(3),$ and $p(4)$ are perfect squares but $p(5)$ is not.

  2. Find a/all quadratic polynomial with integer coefficients $p(x)=ax^2+bx+c$ such that $p(1),p(2),p(3),\ldots, p(n-1)$ are perfect squares but $p(n)$ is not.

Jaakko Seppälä
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See the paper, Gonzalez-Jimenez and Xarles, On symmetric square values of quadratic polynomials, Acta Arithmetica 149.2 (2011) 145-159. From the introduction:

In this note we are dealing with the following problem. Given a degree two polynomial $f (x) = ax^2 + bx + c$ in ${\bf Z}[x]$ which is not a square of a degree one polynomial, how many consecutive integer values $f(i)$ can be squares in ${\bf Z}$? This problem has been considered by D. Allison in 1 and [2], who found infinitely many examples with eight consecutive values, and by A. Bremner in [3], who found more examples with seven consecutive values.

Gerry Myerson
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    It seems worth noting that the linked-to result concerns quadratics where the consecutive squares are symmetric, so that the first is equal to the last (or am I misreading something?), so it doesn't fully answer the OP's question. See also https://math.stackexchange.com/questions/338037/find-a-polynomial-as-2836x2-12724x16129 which asks more or less the same question. – Barry Cipra Jul 21 '19 at 11:53