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It's given ($p$ is a prime): $ x^2-dy^2 \equiv 1\pmod p $ Using only this can we say $ x^2-dy^2 = 1 $ has always integral solution?

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    I do not understand the question. If it is from a book or notes, perhaps you could give more detail.2012-06-28

1 Answers 1

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If $=1 \pmod p$ actually means $\equiv \pmod p$,

$\implies x^2-1 \equiv dy^2 \pmod p$

Observe that if $p$ |LHS, $x≡±1 \pmod p$, then $p|dy^2$.

  1. if $p∤d$ i.e., $(p,d)=1, p|y$ for accepting solution.
  2. if $p|d$, any integral $y$ will give us integral solution.