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How to solve congruence $x^2-2 \equiv 0\pmod a$, $x$ and $a$ are integers, and $a$ mustn't be prime? I have found solution when a is prime, but I haven't found solution for general case.

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  • $\begingroup$ Find solutions modulo all the distinct (highest) prime powers $p^n$ dividing $a$, and use the Chinese remainder theorem to assemble them to a solution modulo $a$. $\endgroup$ – Daniel Fischer Aug 31 '13 at 12:37
  • $\begingroup$ "I have found solution when $a$ is prime" Oh? What's the solution when $a=3$? $\endgroup$ – Gerry Myerson Aug 31 '13 at 12:40
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Building on the comments, a strategy might be the following.

  1. Check that the congruence has a solution modulo all prime divisors $p$ of $a$. If there is no solution even for a single such prime $p$, the congruence itself has no solution. For this, you may use quadratic reciprocity.
  2. Assuming there are solutions for each such $p$, find them, and use Hensel lifting to get a solution modulo each prime power $p^{e}$ such that $p^{e}$ divides $a$, but $p^{e+1}$ does not.
  3. Use the Chinese Remainder Theorem to glue all these solutions together.
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  • $\begingroup$ Sorry, I forgot to say, I can't factorise $a$,because it is very large. $\endgroup$ – Marcin Augustynowicz Aug 31 '13 at 12:59

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