On symmetric square values of quadratic polynomials
In this note we are dealing with the following problem. Given a degree two polynomial f(x) = ax2 +bx+c ∈ Z[x] which is not a square of a degree one polynomial, how many consecutive integer values f(i) can be squares in Z? This problem has been considered by D. Allison in [1] and [2], who found infin...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2011 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/711180 |
| Acesso em linha: | http://hdl.handle.net/10486/711180 https://dx.doi.org/10.4064/aa149-2-4 |
| Access Level: | acceso abierto |
| Palavra-chave: | Elliptic Chabauty Squares Covering Collections Quadratic Polynomials Matemáticas |
| Resumo: | In this note we are dealing with the following problem. Given a degree two polynomial f(x) = ax2 +bx+c ∈ Z[x] which is not a square of a degree one polynomial, how many consecutive integer values f(i) can be squares in 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. The examples found by Allison are all by polynomials which are symmetric with an axis of symmetry midway between two integers. This means that, after some easy translation, all the examples are of the form f(x) = a(x 2 +x) +c and the values are f(i) for i = −3, −2, −1, 0, 1, 2, 3 and 4. This result was obtained by translating the problem to computing rational points on some elliptic curve which has rank one. On the other hand, Bremner [3] shows that there does not exist any example which is symmetric about an integral value and with seven values, by showing that these examples would be described by rational points on some rank zero elliptic curve, which has 12 points, all corresponding to the polynomial f(x) being the square of a polynomial. In the same paper, Bremner asks if there are examples as the ones found by Allison, but with ten consecutive squares. The problem translates to finding all the rational points of a genus 5 curve, a fact already noticed by Allison and by Bremner. He conjectures that there is no such example. In this note we prove this conjecture, and so, together with the results of Bremner and Allison, we get the following theorem |
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