SOME PROBLEMS IN NUMBER THEORY

1. Find all positive integers n such that n²+ 1 is divisible by n+ 1.

2. Find all integers x(x is not equal to 3) such that x-3|x³-3.

3. Prove that there exists infinitely many positive integers n such that

4n²+ 1 is divisible both by 5 and 13.

4. Prove that for positive integer n we have 169|3³ⁿ⁺³-26n-27.

S. Prove that 19|2^26k+2 +3 for k = 0, 1, 2, ....

6. Prove the theorem, due to Kraitchik, asserting that 13|2⁷⁰+3⁷⁰.

7. Prove that 11*31*61|20¹⁵-1.

8. Prove that for every positive integer n the number 3(1⁵+2⁵+ ... +n⁵)

is divisible by 1³+2³+ ... +n³•

9. Find all integers n > 1 such that 1 ⁿ+2ⁿ+ ... +(n-l)ⁿ is divisible

by n.

10.Prove that for positive integer n we have n²|(n+l)ⁿ-1.