what is the set of all positive integers satisfying the equality $2(n^2 - 1)=t(t+1)$ other than $n=t=2$ or $n=4$, $t=5$?
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$\begingroup$ Where did you come across this question, and why these particular parameters? $\endgroup$– Yemon ChoiFeb 10, 2013 at 6:35
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$\begingroup$ Dammit Abhinav, you beat me by 4 minutes! :) $\endgroup$– Greg MartinFeb 10, 2013 at 7:23
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1 Answer
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You can rewrite it as $(2t + 1)^2 = 8n^2 - 7$, so it's essentially equivalent to finding solution for the Pell equation $x^2 - 8y^2 = -7$. This has one solution $(1,1)$, and so it certainly has infinitely many. For example, the solution $(x,y) = (379, 134)$ gives you $n = 134, t = 189$. See, for instance, "An introduction to the theory of numbers" by Niven, Zuckerman and Montgomery, Section 7.8.
