NUMBER-THEORY EXERCISES, IV

DAVID PIERCE

Exercise 1. Prove that the following are equivalent:

(a) Every even integer greater than 2 is the sum of two primes.

(b) Every integer greater than 5 is the sum of three primes. Exercise 2. Infinitely many primes are congruent to −1 modulo 6. Exercise 3. Find all n such that

(a) n! is square;

(b) n! + (n + 1)! + (n + 2)! is square. Exercise 4. Determine whether a^{2} ≡ b^{2} (mod n) ⟹ a ≡ b (mod n). Exercise 5. Compute ∑_{k=1}^{1001} k^{365} (mod 5). Exercise 6. 39 | 53^{103} + 103^{53}. Exercise 7. Solve 6^{n+2} + 7^{2n+1} ≡ x (mod 43). Exercise 8. Determine whether a ≡ b (mod n) ⟹ c^{a} ≡ c^{b} (mod n). Exercise 9. Determine r such that a ≡ b (mod r) whenever a ≡ b (mod m) and a ≡ b (mod n). Exercise 10. Solve the system

x ≡ 1 (mod 17),
x ≡ 8 (mod 19),
x ≡ 16 (mod 21).

Exercise 11. The system

\begin{cases}x ≡ a mod n \\ x ≡ b mod m \end{cases}

has a solution if and only if gcd(n, m) | b − a.

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 10, 2007.