MA-410, Spring 2023, Homework 2,
due as indicated for each problem.
All solutions must be submitted on the Moodle web site
for the class at
wolfware.ncsu.edu.
You may upload a photo of your handwritten solution or
a file of your typed solution.
Note my office hours on my
schedule.
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Due Friday, March 3, 11:59pm
Please prove for all positive integers M, N that
GCD(2M – 1,
2N – 1)
= 2GCD(M, N) – 1.
-
Due Friday, March 3, 11:59pm
Let the Fermat numbers be Fn= 22n + 1
for integers n ≥ 0. Please prove that
2Fn-1
≡ 1 (mod Fn) for all n ≥ 0.
[cf. ENT, §5.2, Problem 15(b), page 93.]
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Due Thursday, March 9, 11:59pm
Using the Chinese Remainder Algorithm from class,
which is based on interpolation by divided differences,
compute an integer N such that
25 = 52 divides N,
16 = 24 divides N+1,
and
27 = 33 divides N+2.
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Due Thursday, March 9, 11:59pm
ENT, §5.2, Problem 13, page 93.
Assume that p and q are distinct odd primes such that
(p – 1)
divides
(q – 1). If GCD(a, pq) = 1, show that
aq – 1 ≡ 1 (mod pq).
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Due Thursday, March 9, 11:59pm
ENT, §5.2, Problem 19, page 93.
Prove that if 6k+1, 12k+1 and 18k+1 are all prime numbers,
then (6k+1)(12k+1)(18k+1) is an absolute pseudoprime. Example:
1729 = 7 times 13 times 19.