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...
Before reading the answer can I interest you in a clue?
The solution relies on showing that p2 - 1 is a multiple of 2x2x2x3
First expand p2 - 1 to give:
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:
* - I know 2 is prime, and even. But we are in the space greater than 3
I think they both got this one right.
If you're curious what Bard made of this puzzle...
If you're curious what ChatGPT made of this puzzle...
The solution relies on showing that p2 - 1 is a multiple of 2x2x2x3
First expand p2 - 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:
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
I think they both got this one right.
If you're curious what Bard made of this puzzle...
If you're curious what ChatGPT made of this puzzle...
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