MA-410 Homework 2

Due at 4:59pm in my mailbox in SAS 3151, Thursday, March 2, 2017


All solutions must be submitted stapled in hardcopy either to me in class or placed in my mailbox.
Note my office hours on my schedule.

  1. A Mersenne number is an integer of the form Mp = 2p - 1, where p is a prime number. Note that for p = 11; M11 = 2047 is divisible by 23 and 89. Please prove that no Mp is divisible by 11.
  2. Please prove that there are infinitely many primes that are congruent 5 (modulo 6). [Hint: Suppose there are finitely many. Let Q be the product of all such primes. Consider the prime factors of N = 3Q + 2. Take all modulo 6.
  3. ENT, §4.2, Problem 6(d), page 68.
  4. ENT, §4.4, Problem 18, page 83, using the Chinese remainder algorithm (based on Newton interpolation) from class. [Hint: separate the cases x ≡ 2 (mod 6) and x ≡ 5 (mod 6).]
  5. ENT, §5.2, Problem 12, page 93.
  6. ENT, §5.2, Problem 19, page 93.