MA-410 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.

1. Due Thursday, March 11, 11:55pm
   A Mersenne number is an integer of the form M_p = 2^p - 1, where p is a prime number. Note that for p = 11, M₁₁ = 2047 is divisible by 23 and 89. Please prove that no other Mersenne number is divisible by 23. [Hint: compute 2^2k+1 mod 23 for k=1,2,3,... by Maple/wolframalpha.]
2. Due Thursday, March 11, 11:55pm
   Let the Fermat numbers be F_n= 2²ⁿ + 1 for integers n ≥ 0. Please prove that 2^F_n-1 ≡ 1 (mod F_n) for all n ≥ 0. [cf. ENT, §5.2, Problem 15(b), page 93.]
3. Due Tuesday, March 16, 11:55pm
   Using the Chinese remainder algorithm from class, which is based on interpolation by divided differences, compute an integer N such that 16 = 2⁴ divides N, 27 = 3³ divides N+1 and 25 = 5² divides N+2.
4. Due Tuesday, March 16, 11:55pm
   ENT, §5.2, Problem 14, page 93. If p and q are distinct prime numbers, prove that p^q-1 + q^p-1 ≡ 1 (mod pq).
5. Due Thursday, March 18, 11:55pm
   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.