Steve Hayman 🇨🇦🇬🇱TIL: For any prime number p >= 5, p²-1 is divisible by 24.
That's cool.
Isn't it?
Yes it is.
Informal proof that for a prime p >=5, p²-1 must be a multiple of 24.
p²-1 = (p-1)(p+1)
p-1, p, p+1 are three consecutive integers. One of them must be divisible by 3 - and it can't be p, because p is prime. So either p-1 or p+1 is a multiple of 3.
Also, p is odd, so p-1 and p+1 are both even - and one or the other must be divisible by 4. One is a multiple of 2, the other of 4.
So the product of p-1 and p+1 has factors of 2, 3 and 4, and must be a multiple of 2*3*4 =24.
iNecas
Barry Schwartz 🫖So often things are so once properly understood. They start to seem trivial. But that means the logic is sinking in!
Gödel’s theorem turned out to be clickbait, for instance. Because an inconsistent system of any kind could always have ‘proven’ itself ‘consistent’, there was never any point in trying to come up with a consistent formal system that could prove itself consistent! The whole effort was a waste of human life hours.
But only now do we see that this is so.