Pries: 405 Number Theory, Spring 2012. Homework 1.
Due: Friday 1/27.
Primes Read: Stein chapter 1.
Problems:
Do 8 of the following problems.
1. Stein 1.3, 1.5 (s), 1.7 (s), 1.8b, 1.13.
2. The odd numbers 3,5,7 form a prime triplet. Explain why this is the only prime triplet.
In other words, if p, p + 2, and p + 4 are all prime, prove that p = 3.
3. Do you think there are infinitely many primes of the form p = N 2 − 1? Do you think
there are infinitely many primes of the form p = N 2 − 2?
4. Calculate the sums P
1, 1 + 3, 1 + 3 + 5 + 7, 1 + 3 + 5 + 7 + 9. What pattern do you see?
Find a formula for ri=1 (2i − 1). Use induction to prove your formula is correct.
5. Which primes can be written as the sum of three squares? State a conjecture and
prove one direction of it.
6. Prove that the greatest common divisor of any two consecutive terms of the Fibonacci
sequence 1, 1, 2, 3, 5, 8, . . . equals 1.
7. Find a solution in integers to the equation:
12345x + 67890y = gcd(12345, 67890).
Thought is only a flash between two long nights, but this flash is everything.
- Henri Poincare
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