332 MATH - ALGEBRA AND NUMBER THEORY : HOMEWORK 6
DUE FRIDAY, OCTOBER 5TH (NO LATER THAN 12PM)
NOTE: Homework sets 5 and 6 will be due on October 5th. Homework 6 Chapter 3: The theorems of Fermat, Wilson and Euler Solve the following exercises from Adler & Coury: Exercises (pages 94, 95): 33 Also solve the following problems: Problem 1. Prove that a12 - 1 is divisible by 35 for all integers a relatively prime to 35. Also, find the remainder of 512 - 1 and 712 - 1 modulo 35. Problem 2. Use congruences to find integers a, b, c such that 10886664 = a 4! + b 5! + c 10! and 0 a, b, c 3 (hint: 10886664 5 mod 7). A trial and error solution will earn no points. Problem 3. Write down all quadratic residues and quadratic non-residues modulo 5, 7, 11 and 13. Also, verify that if a and b are quadratic non-residues modulo 13 then a b is a quadratic residue modulo Problem 13. 4. Calculate the following values of the Legendre symbol, using the definition or Euler's criterion: 21 5 7 280 381 3 , , , , , . 7 7 5 41 41 17 Problem 5. Let p be a prime. Prove that there are exactly p and p-1 quadratic non-residues. 2
p-1 2
quadratic residues modulo
Problem 6. Use Euler's criterion to prove that the product of two quadratic residues is a quadratic residue modulo p (odd prime), and that the product of two quadratic non-residues modulo p is a quadratic residue modulo p. Problem 7....