On symmetric square values of quadratic polynomials
Enrique Gonzalez-Jimenez, Xavier Xarles
TL;DR
This paper investigates quadratic polynomials with integer coefficients and symmetry axes, proving the non-existence of certain square value sequences for specific lengths, and establishing infinite cases for others.
Contribution
It establishes new bounds on the length of consecutive square values for symmetric quadratic polynomials with integer coefficients.
Findings
No non-square quadratic polynomial with integer coefficients and symmetry axis takes 7 or more consecutive square values.
Infinite such polynomials exist when N <= 6 or N=8.
The results delineate precise conditions for the occurrence of consecutive square values.
Abstract
We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N=7 or N >= 9. At the opposite, if N <= 6 or N=8 there are infinitely many.
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