Let $p=4k+1$ be a prime number such that $p=a^2+b^2$, where $a$ is an odd integer.Prove that the equation $$x^2-py^2=a$$ has at least a solution in $\mathbb{Z}$.
Pell type equation: $x^2-py^2=a$
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number-theory
algebraic-number-theory
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0A standard way to find $a,b$ begins by expanding $\sqrt p$ in a continued fraction; if you stop halfway through the period of the continued fraction, somehow you get $a$ and $b$. I bet solving $x^2-py^2=a$ is related to this. Sorry I can't be more precise. – 2012-08-05
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1$p=13$, $a=3$, $x=4$, $y=1$. – 2012-08-06
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0@Sil, I think the idea was to prove that for *every* such $p$ there's at least one solution. – 2012-08-06
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0@GerryMyerson, Ah, in my defence, that wasn't made very clear. – 2012-08-06
1 Answers
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See e.g. Gary Walsh, On a question of Kaplansky".
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0The Albanian Journal of Mathematics! – 2012-08-06
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0Thank you, Franz. I remember Kap corresponding with Mollin about a bunch of conjectures, Mollin sent a short pdf with a proof of one of them, I had not realized Mollin published them(2001). Campus has C.R...Canada in bound volumes, but no scan and post service in this case, I will need to go borrow the actual volume and do my own copying. – 2012-08-06
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0Abstract here: http://mr.math.ca/Vol_23/No_2.html – 2012-08-07