A neat twin primes observation

Take a twin prime pair ((except for 3 and 5)) and sum the squares of its two numbers. The result is two more than a multiple of 72.

Twin primes are primes that differ in value by two – such as (11 and 13) or (71 and 73). It’s a long-standing open problem whether there are infinitely many of them – the late Vicky Neale’s Closing The Gap is a brilliant book if you want to know more.

It’s also a brilliant book if you don’t want to know more, but you’d probably find it less interesting in that case.

I digress. You might want to have a play with this and see if you can prove it yourself before I do – with spoilers, as always, below the line.

Matt Parker has a thing about 2 and 3 not being real primes, but sub-primes – they’re only primes by virtue of being so small, and they’re not really proper primes. One of the reasons for this is that there’s something nice about “proper” primes: they’re all either one more or one less than a multiple of six.

Why’s that? Well, numbers two more or less than a multiple of six are even. Numbers three more and less than a multiple of six are threeven (that is, multiples of three). The only remaining possibilities are one more and one less.

For twin primes in particular, the only possibility is for them to be either side of a multiple of six – one is one more and the other is one less. (The only alternative is for one of them to be a multiple of three or them both to be even, which would mean not proper primes.)

In that case, your twin primes can be written as $6k -1$ and $6k+1$, so their squares are $36k^2 - 12k + 1$ and $36k^2 + 12k + 1$. Adding them together, the $k$ terms vanish and you’re left with $72k^2 + 2$, which is clearly two more than a multiple of 72.

I found it surprising (although explicable) that 72 should crop up – it seems like a very big number to be involved here!