If a is a quadratic nonresidue of the odd primes p and q, then is the congruence [itex] x^2 \equiv a (\text{mod } pq) [/itex] solvable?(adsbygoogle = window.adsbygoogle || []).push({});

Obviously, we want to evaluate [itex] \left( \frac{a}{pq} \right) [/itex]. I factored a into its prime factors and used the law of QR and Euler's Criterion to get rid of the legendre symbols needed to evaluate [itex]\left( \frac{a}{pq}\right)[/itex]. I don't believe that this helped, though, because I get that it is conditionally solvable, which I don't think is possible from the way the question is worded. (To be exact, I concluded that if a has only one prime factor, then it is unsolvable unless it is 2. It is solvable for every other case.)

Any help is appreciated.

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# Quadratic Reciprocity

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