**This page has a few puzzles involving prime numbers:**

1. How many of the three-digit numbers that can be made from all of the the digits 1, 3 and 5 (used only once each) are prime?

2. 12345 can be expressed as the sum of two primes in exactly one way. What is the larger of the two primes whose sum is 12345?

3a. Find all prime numbers p such that 2p+1 is a perfect square.

3b. Find all prime numbers p such that 2p+1 is a perfect cube.

4a. Let n be an integer greater than 6. Prove that if n-1 and n+1 are both prime, then n²(n²+16) is divisible by 720.

4b. Is the converse true? That is, if n²(n²+16) is divisible by 720, then are n-1 and n+1 both prime?

5. Let a be the integer whose base 10 representation consists of 119 ones. Prove that a is not prime.

6. Prove composite: n^{4}+4^{n}, n>1.

Source: various sources