Math 345/645 - Weekly homework 6
The problems on this assignment should be written up impeccably and turned in on Tuesday,
March 22. This assignment needs to be typed in LATEX. A problem with a ∗ is extra-credit for
undergraduates and required for graduate students.
1. Let n be a positive integer and suppose that a ∈ Z with gcd(a, n) = 1 and ordn (a) = n − 1.
Prove that n is prime. [ This fact is the main tool used to prove that large numbers are prime.
]
2. Let p be a prime number and Mp = 2p − 1. Let q be a prime divisor of Mp . Prove that
q ≡ 1 (mod p). (This can be used to find infinitely many primes by setting p1 = 2, and pn a
prime divisor of 2pn−1 − 1 for n ≥ 2.)
3. ∗ Prove that if n > 1 is a positive integer, then n - 2n − 1. (Hints for this problem are
available on the class diary.)
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