Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p reduction properties of the Legendre symbol (a|p) in conjunction with the Law of Quadratic Reciprocity and the two commonly given special cases derived from Euler's Criterion, namely that (-1|p) = +1 iff p==1 (mod 4) and (2|p) = +1 iff p == +- 1 (mod 8) for odd primes p.

Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p reduction properties of the Legendre symbol (a|p) in conjunction with the Law of Quadratic Reciprocity and the two commonly given special cases derived from Euler's Criterion, namely that (-1|p) = +1 iff p==1 (mod 4) and (2|p) = +1 iff p == +- 1 (mod 8) for odd primes p.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 46E: Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection...

See similar textbooks

Similar questions

30. Prove statement of Theorem : for all integers .
a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3
Show that the converse of Eisenstein’s Irreducibility Criterion is not true by finding an irreducible such that there is no that satisfies the hypothesis of Eisenstein’s Irreducibility Criterion.
Prove that if n is a positive integer greater than 1 such that n is not a prime, then n has a divisor d such that 1dn.