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 2005-09-11, 10:47 #5 cyrix   Jul 2003 Thuringia; Germany 2×29 Posts Hi T.Rex! For prime $M_q$ with $q \equiv 1 \pmod 4$ (and q>1) your conjecture is true: Because of the quadratic reciprocity law (Gauß) 5 is a quadratic residue of $M_q$, iff $M_q$ is quadratic residue of 5 (because both are of the from 4n+1). But $M_q=2^q-1=2^{4n+1}-1 \equiv 2^1-1=1^2 \pmod 5$ so 5 is a quadratic residue of $M_q$ and there existists two numbers X and Y, which have the properties $X \equiv -Y \pmod {M_q}$ and $x^2 \equiv y^2 \equiv 5 \pmod {M_q}$ Yours, Cyrix