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How might one show that if given that there exist some integer $x_0$ s.t. ${x_0}^2 \equiv c \pmod p$ for some integer $c$ that is not a square and not a multiple of $p$, then there exist $x_n$ s.t. $x_n \equiv x_{n+1} \pmod p$ & $x_{n+1} \equiv c \pmod p^{n+1}$? Thanks in advance.

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Your question is unclear. If $x_n=c$ for all $n$ then your conditions are satisfied (for all $p$, in fact). Please give some thought to what you really want to ask. –  Gerry Myerson Aug 15 '11 at 12:52
Perhaps you will find helpful Hensel's Lemma, though, perhaps, your question - once fixed - may be a special case. –  Bill Dubuque Aug 15 '11 at 13:43
@Gerry Myerson: Thanks for spotting the mistake. The question should make more sense now. ;) –  Ola Aug 15 '11 at 14:46
@Bill Dubuque: Thanks for the suggestion, it looks promising. I am processing the article... :) –  Ola Aug 15 '11 at 14:51
Ola, part of the question is fixed, part is still broken. Don't you want the condition to be on $x_{n+1}^2$? –  Gerry Myerson Aug 15 '11 at 22:48
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