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5. Using the Quadratic Reciprocity Law, prove that P 1 if p=1 or 11 (mod 12), 1 if p = 5 or 7 (mod 12) -- ifp=5 for each odd prime p.

5. Using the Quadratic Reciprocity Law, prove that P 1 if p=1 or 11 (mod 12), 1 if p = 5 or 7 (mod 12) -- ifp=5 for each odd prime p.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 27E
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5. Using the Quadratic Reciprocity Law, prove that 1 if p=1 or 11 (mod 12), -- ifp-5 or 7 (mod 12) for each odd prime p. THEOREM 9-4 (Quadratic Reciprocity Law): If p and q are distinct odd primes, then which case (2)+(2) = unless p= q = 3 (mod 4), in REMARK: This theorem may be stated equivalently in the form = ()()(-1)- (p-1)(q-1)/4
Transcribed Image Text:5. Using the Quadratic Reciprocity Law, prove that 1 if p=1 or 11 (mod 12), -- ifp-5 or 7 (mod 12) for each odd prime p. THEOREM 9-4 (Quadratic Reciprocity Law): If p and q are distinct odd primes, then which case (2)+(2) = unless p= q = 3 (mod 4), in REMARK: This theorem may be stated equivalently in the form = ()()(-1)- (p-1)(q-1)/4
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