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 nē(nē+16) is divisible by 720.

4b. Is the converse true? That is, if nē(nē+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: n⁴+4ⁿ, n>1.

Partial credit is given for partial solutions, and extra credit is given for clever, imaginative, or especially detailed solutions.

Source: various sources

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Let's play a game. I'll be the computer, and you be the human, OK? I'll hand you a slip of paper on which will be printed two numbers -- all numbers are nonnegative integers. Then you can reduce one or both of the two numbers by any positive integer amount, but you can't make either number negative, and then you can hand me back the slip of paper. We take turns reducing one or both of the numbers until one of us reduces both numbers to 0, winning the game.

What is a good strategy for winning the game?

Would you like to play the game while you think some more about it?

Source: reference will be supplied at a later date.

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