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MATH 2601 - FoMP
Problem 1. Complete the following proof (by contradiction) that there are infinitely many primes of the form 4k + 3.
Suppose 3 = p1, 7 = p2, p3, . . . , pn are all the finitely many primes of the form 4k + 3. Then consider M = (p1p2p3 · · · pn) 2 + 2.
i) Argue that M is of the form 4k + 3.
ii) Argue that none of the primes pi divides M.
iii) Complete the proof by arriving at a contradiction.
Problem 2. Prove that there are infinitely many primes of the form 3k + 2.
Problem 3.
i) Compute g := GCD(561, 25) using Euclid’s algorithm.
ii) Compute integers x and y so that 561x + 25y = g.
Reading Exercise. Read Section 1.1 from Hammack’s book.
Additionally, turn in (the solutions to) the following problems from Hammack’s book.
Sec.1.1: 16, 22, 28 Sec.1.2: 18, 20