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Number Theory

Number Theory. By: Jarek Esswein. 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)

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Number Theory

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  1. Number Theory By: Jarek Esswein

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

  3. 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)

  4. Solving the problem

  5. 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.

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