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Sharp bounds on the number of solutions of $X^{2}-\left( a^{2}+b^{2} \right) Y^{4}=-b^{2}$
- Publication Year :
- 2018
-
Abstract
- We generalise and improve a result of Stoll, Walsh and Yuan by showing that there are at most two solutions in coprime positive integers of the equation in the title when $b=p^{m}$ where $m$ is a non-negative integer, $p$ is prime, $(a,p)=1$, $a^{2}+p^{2m}$ not a perfect square and $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has an integer solution. This result is best possible. We also obtain best possible results for all positive integer solutions when $m=1$ and $2$. When $b$ is an arbitrary square with $(a,b)=1$ and $a^{2}+b^{2}$ not a perfect square, we are able to prove there are at most three solutions in coprime positive integers provided $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution and $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-b^{2}$ has only one family of solutions. Our proof is based on a novel use of the hypergeometric method that may also be useful for other problems.<br />first draft for CNTA 2018. Comments welcome!
- Subjects :
- Mathematics - Number Theory
FOS: Mathematics
Number Theory (math.NT)
11D25, 11J82
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....7fd2e222d51f2739cad792c2b2caae8c