Infinitely Many Quadratic Diophantine Equations Solvable Everywhere Locally , But Not Solvable Globally ∗

@inproceedings{Mollin2004InfinitelyMQ,
  title={Infinitely Many Quadratic Diophantine Equations Solvable Everywhere Locally , But Not Solvable Globally ∗},
  author={Richard A. Mollin},
  year={2004}
}
We present an infinite class of integers 2c, which turn out to be Richaud-Degert type radicands, for which 2x2 − cy2 = −1 has no integer solutions, but for which 2x2 − cy2 ≡ −1 (mod n) has integer solutions for any n ∈ N. This explains a topic often seen in introductory number theory courses, that appears as a curiosity, yet for which we are able to give the underlying reasons in terms of simple continued fraction expansions. 

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