sol6 - Math 465 Solution Set 6 Ae Ja Yee 1. Evaluate the...

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Unformatted text preview: Math 465 Solution Set 6 Ae Ja Yee 1. Evaluate the following Legendre symbols. (Note that 127 is prime.) (a) ( 2 127 ) (b) (- 1 127 ) (c) ( 5 127 ) (d) ( 15 127 ) Solution. (a) 127 ≡ - 1 (mod 8) , 2 127 = 1 (b) 127 ≡ 3 (mod 4) ,- 1 127 =- 1 (c) 5 127 = (- 1) 2 · 63 127 5 = 2 5 =- 1 (d) 15 127 = 3 127 5 127 = (- 1) 1 · 63 127 3 (- 1) 2 · 63 127 5 = 1 3 = 1 2 2. How many solutions do the following congruences have? (a) x 2 ≡ 2 (mod 61) (b) x 2 ≡ - 2 (mod 59) (c) x 2 + 5 x + 1 ≡ 0 (mod 13) Solution. (a) Since 61 ≡ 5 (mod 8), 2 61 =- 1 , so no solution exists. (b) Since 59 ≡ 3 (mod 4) and 59 ≡ 3 (mod 8)- 2 61 =- 1 61 2 61 = (- 1)(- 1) = 1 , so there exist solutions. If x is a solution to the congruence, then so is- x , which is not congruent to x , so there are two distinct solutions. 1 (c) x 2 + 5 x + 1 ≡ x 2- 8 x + 1 ≡ ( x- 4) 2- 2 ≡ (mod 13) , which is equivalent to ( x- 4) 2 ≡ 2 (mod 13). Since 13 ≡ 5 (mod 8), ( 2 13 ) =- 1. This shows that no solution exists.shows that no solution exists....
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This note was uploaded on 07/23/2008 for the course MATH 465 taught by Professor Yee during the Spring '08 term at Penn State.

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sol6 - Math 465 Solution Set 6 Ae Ja Yee 1. Evaluate the...

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