Answer to Puzzle #15: Prime Squared Minus 1 Multiple of 24

15. Why is it that if 'p' is a prime number bigger than 3, then p2-1 is always divisible by 24 with no remainder?

This one took me a while, my first answer, although valid, was not the best. The final answer is quite tidy although there is some algebra required...





The solution relies on showing that p2 - 1 is a multiple of 2x2x2x3



First expand p2 - 1 to give: Before reading the answer can I interest you in a clue? The solution relies on showing that p- 1 is a multiple of 2x2x2x3First expand p- 1 to give: p2 - 1 = (p - 1) x (p + 1) Then consider the terms on the right hand side, firstly we know that p must be odd (no even prime numbers *,) so p - 1 and p + 1 must be even. We have two of the factors we require.



Additionally since p - 1 and p + 1 effectively form 2 consecutive even numbers one of them must be a multiple of 4, thus we have another of our factors of 2. So far we have 2x2x2, now to get the factor of 3



p - 1, p & p + 1 form three consecutive numbers. In any three consecutive numbers one will be a multiple of 3, we know it is not p which is a multiple of 3, as this is prime, hence either p - 1 or p + 1 is a multiple. Therefore p2 - 1 has the factors 2, 2, 2 & 3 hence: Then consider the terms on the right hand side, firstly we know that p must be odd (no even prime numbers,) so p - 1 and p + 1 must be even. We have two of the factors we require.Additionally since p - 1 and p + 1 effectively form 2 consecutive even numbers one of them must be a multiple of 4, thus we have another of our factors of 2. So far we have 2x2x2, now to get the factor of 3p - 1, p & p + 1 form three consecutive numbers. In any three consecutive numbers one will be a multiple of 3, we know it is not p which is a multiple of 3, as this is prime, hence either p - 1 or p + 1 is a multiple. Therefore p- 1 has the factors 2, 2, 2 & 3 hence: p2 - 1 = 24n (Why must p be greater than 3? Well 3 is the only number which is both a multiple of 3 and prime)

* - I know 2 is prime, and even. But we are in the space greater than 3





Where next?

List of Puzzles Random Puzzle Next Puzzle Where next?



(Why must p be greater than 3? Well 3 is the only number which is both a multiple of 3 and prime)* - I know 2 is prime, and even. But we are in the space greater than 3

© Nigel Coldwell 2004 - – Theon this site may be reproduced without further permission, I do not claim copyright over them. Theare mine and may not be reproduced without my expressed prior consent. Please inquire using the link at the top of the page. Secure version of this page