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I want to calculate wether $\exists x : x^2 \equiv 123 \mod 11\cdot 13$ or not. I do know that in terms of the legendre symbol follows that $\neg(\exists x: x^2 \equiv 123 \mod 11)$ and $\neg(\exists x: x^2 \equiv 123 \mod 13)$. How can i deduce from that, that $\neg(\exists x: x^2 \equiv 123 \mod 11 \cdot 13)$ ? The intention is that the values of the legende-symbol and the jacobi-symbol do not have to be equal.

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    @André Your last sentence puzzles me: the Jacobi symbol extends the Legendre symbol, so they agree whenever both are defined. I think what you're trying to say is that the Jacobi symbol can't be relied upon to identify quadratic residues modulo composite bases.2012-11-15

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Assume there exists $x$ such that $x^2 \equiv 123 \pmod{11\cdot 13}$. Then, by definition, $x^2 = 123 + 11\cdot 13\cdot k = 123 + 11(13k)$ which implies that $x^2 \equiv 123 \pmod{11}$. Contradiction.