BY: JAREK ESSWEIN Number Theory. Problem Use the quadratic law of reciprocity to compute the following Legendre symbol (85/101)

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Presentation transcript:

BY: JAREK ESSWEIN Number Theory

Problem Use the quadratic law of reciprocity to compute the following Legendre symbol (85/101)

Legendre symbol/ Law of Quadratic Reciprocity (-1/p)  1 if p=1 (mod 4)(QR)  -1 if p=-1 (mod 4)(NR) (2/p)  1 if p=1 or 7 (mod 8)(QR)  -1 if p=3 or 5 (mod 8)(NR) (q/p)  (p/q) if p or q= 1 (mod 4)  -(p/q) if p=q=3 (mod 4)

Solving the problem (1)^4*(1)=1 ((2/17))^4*((1/5)=1) (101/17)*(101/5) (17/101)*(5/101) (85/101)

Answer Legendre symbol of (85/101)=1  Implies that 85 is a quadratic residue of 101 and x^2=85(mod 101) has a solution.