Algebraic Manipulation Worksheets With Answers - Serge Ballif Page 3
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1
2
3Serge Ballif
MATH 567 Homework 1
Tuesday, September 2, 2008
Suppose that there are finitely many primes p
of the form 4k 3. Then
p
p
1
2
n
consider the number m
4(p
1)
3
4p
1. None of the p
can
p
p
p
1
2
n
1
n
i
divide m, so by the first part of this exercise, m must have another factor of the form
4k 3 contradicting the assumption that p
p
p
are the only primes of this form.
1
2
n
Similarly, if we assume that q
q
q
are the only primes of the form 6k
5, then
1
2
n
6q
q
q
1 would be a number of the form 6k
5 without a factor of the same
1
2
n
form. Hence there must be infinitely many primes of the form 6k
5.
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