how many consecutive integers $x$ can make $ax^2+bx+c$ square ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:37:54Z http://mathoverflow.net/feeds/question/88233 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88233/how-many-consecutive-integers-x-can-make-ax2bxc-square how many consecutive integers $x$ can make $ax^2+bx+c$ square ? spanferkel 2012-02-11T22:17:47Z 2012-02-12T04:38:05Z <p>The following problem was raised in <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&amp;t=374534" rel="nofollow">a Mathlinks thread</a>: </p> <p><strong>If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?</strong> </p> <p>The polynomial $-15x^2+64$ is obviously a square for the five numbers $x=-2,...,2$, but the method used for finding this in the above thread cannot be extended further. <strong>Should $5$ really be the best possible answer?</strong></p> <p>Has this problem been treated somewhere else?</p> http://mathoverflow.net/questions/88233/how-many-consecutive-integers-x-can-make-ax2bxc-square/88245#88245 Answer by GH for how many consecutive integers $x$ can make $ax^2+bx+c$ square ? GH 2012-02-11T23:43:01Z 2012-02-12T04:38:05Z <p>To resonate with Noam Elkies' comments, it is conjectured that $8$ squares is the maximum for arbitrary $a$, and $4$ squares is the maximum for $a=1$. For $5$ <em>symmetric</em> squares the smallest known leading coefficients are $a=15$ and $a=-20$, while for $5$ <em>increasing</em> squares they are $a=60$ and $a=-56$. It is known that there are infinitely many examples with $5$ or $8$ <em>symmetric</em> squares, or with $6$ <em>increasing</em> squares. It is also known that there is no symmetric sequence of $7$ squares, and only finitely many of $10$ squares up to obvious equivalences (this one follows from Falting's theorem applied to a specific hyperelliptic curve). </p> <p>Good starters are:</p> <p>Browkin-Brzeziński: On sequences of squares with constant second differences, Canad. Math. Bull. 49 (2006), 481–491.</p> <p>Bremner: On square values of quadratics, Acta Arith. 108 (2003), 95–111.</p>