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2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol () in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.

2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol () in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol (9) in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.
Transcribed Image Text:2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all positive integers m, n p(mn) > y(m)p(n) with equality holding only for coprime m and n. (b) State Euler's Criterion and Gauss's Lemma and demonstrate both of them by com- puting the Legendre symbol (9) in two different ways. State the Law of Quadratic Reciprocity and use it to describe all primes p, modulo which a = 17 is a quadratic residue.
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