A prime number is a number whose only factors are 1 and itself. Prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on. A more careful definition is that a prime number is an integer (whole number) greater than 1 whose only factors are 1 and itself.
Question 1: If you take the prime number 5, square it, and subtract 1, you get 24. If you do the same thing to 7, you get 48, which is twice 24. If you try 11, you get 120, which is 5 times 24. Is there a pattern here? Can you show that, for any prime number p that is at least equal to 5, the value of p2-1 is a multiple of 24?
Thanks to Jeff Borggaard for this puzzle.
Question 2: Let Pi indicate the i-th prime number. Let S(m) be the product of the first m prime numbers, that is,
S(m) = P1 P2 P3...Pm. Obviously, S(m) can never be a perfect square. But can S(m)+1 be a perfect square?
I give up, show me the solution.