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Old 2005-09-11, 10:47   #5
Jul 2003
Thuringia; Germany

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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}

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