NUMBER-THEORY EXERCISES, III

DAVID PIERCE

Here p and p_{i} are always prime numbers. Exercise 1. p ≡ ±1 (mod 6). Exercise 2. If p ≡ 1 (mod 3) then p ≡ 1 (mod 6). Exercise 3. If n ≡ 2 (mod 3), then n has a factor p such that p ≡ 2 (mod 3). Exercise 4. Find all primes of the form n^{3} − 1. Exercise 5. Find all p such that 3p + 1 is square. Exercise 6. Find all p such that p^{2} + 2 is prime. Exercise 7. n^{4} + 4 is composite. Exercise 8. If n is positive, then 8^{n} + 1 is composite. Exercise 9. Find all integers n such that the equation

x^{2} = ny^{2}

has only the zero solution. Prove your findings. Exercise 10. If p_{0} < ⋯ < p_{n}, prove that the sum

\frac{1}{p_{0}} + ⋯ + \frac{1}{p_{n}}

is not an integer.

Mathematics Dept, Middle East Technical University, Ankara 06531, Turkey E-mail address: dpierce@metu.edu.tr URL: http://www.math.metu.edu.tr/~dpierce

Date: October 5, 2007.