If $a,b$ are positive integers and $b$ is odd , then is it ever possible that $ \dfrac{2a^2-1}{b^2+2} $ is an integer ?
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Since $b$ is odd, $b^2 + 2\equiv 3\pmod{8}$. Thus $2$ is not a quadratic residue mod $(b^2 + 2)$. Hence $2a^2\not\equiv 1\pmod{b^2 + 2}$, from which the result follows.
anomaly
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1You want to work modulo $8$, note that for example $2$ is a QR of $7$. – André Nicolas Jun 19 '14 at 04:59
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Oops, yes, thanks: (2/b) depends on b mod 8, not 4. – anomaly Jun 19 '14 at 05:00
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In principle the QR business has to do with primes. So one really should say that $b^2+2$ has a prime divisor $p$ of the form $8k\pm 3$. Then work modulo $p$. – André Nicolas Jun 19 '14 at 05:04
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Er, why was this (now year-old) answer voted down? – anomaly Jun 10 '15 at 15:58
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Someone may want to have the question deleted, because OP did not "show work." That is something I do not approve of, since it can serve to deprive the site of interesting questions and/or answers. – André Nicolas Jun 10 '15 at 16:10
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...but a year after the original post? – anomaly Jun 10 '15 at 17:01
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Can this formula to use. How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$? But we'll see this topic and select the desired formula. Families of curves over number fields
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It is difficult to substitute in the formula and check if the root be whole? Why formulas do not like? – individ Jun 19 '14 at 05:06