Math 345/645 - Weekly homework 9
The problems on this assignment should be written up impeccably and turned in on Tuesday,
April 19. This assignment needs to be typed in LATEX. A problem with a ∗ is extra-credit for
undergraduates and required for graduate students. A problem with a ∗∗ is extra-credit for
undergraduates and graduate students.
1. Suppose that p ≡ 3 (mod 4) is a prime number and q = 2p + 1 is also prime. Prove that q
divides 2p − 1.
2. Prove that there are infinitely many primes p ≡ 9 (mod 10) using the following procedure.
(a) Show that there is at least one such prime.
(b) Assume there are finitely many and let P be the product of all such primes. Let N = 4P 2 −5.
Prove that all prime divisors of N are congruent to 1 or 4 mod 5.
(c) Show that N ≡ 9 (mod 10) and conclude that there is a prime p|N so that p ≡ 9 (mod 10).
Use this to derive a contradiction.
3. ∗ Prove
that if a is a positive integer and p and q are primes with p ≡ q (mod 4a), then
a
a
=
. [ Use Gauss’s lemma version 2 and write q = p + 4am. Show that the number of k,
p
q
q−1
1 ≤ k ≤ 2 with b 2ka
c = s is equal to 2m more than the number of k with 1 ≤ k ≤ p−1
with
q
2
2ka
b p c = s. ]
4. ∗∗ (Due the last day of class). Carefully write up proofs of Theorems 7.1 through 7.17.
Write-ups will be graded based on the written homework rubric. The whole collection of
problems will be worth 17 written homework points.
1