19Jul/1011

## Interesting relationship among primes

Many properties of primes are very difficult to determine and prove. Here's an exploration that's within reach of many of us:

What is interesting about the difference of the squares of most any two primes? In other words, what is interesting about p

_{1}^{2}-p_{2}^{2}for most primes p_{1}and p_{2}? When does this property hold? Prove your assertion.

I got the idea for this puzzle/exploration from Standup Maths and adapted it to make it harder!

Tim CowlishawJuly 19th, 2010 - 04:50

I’ve got this so far: http://mathurl.com/34erol5

Will try and work out a proof of it!

Tim CowlishawJuly 19th, 2010 - 04:57

Aah – that can’t be what you’re after as it’s true for all N, not just primes. back to the drawing board!

JonathanJuly 19th, 2010 - 07:21

Can I restrict my observations to the odd primes?

SolJuly 19th, 2010 - 07:43

Jonathan – The observation I’m thinking of is not true for all odd primes, just most of them.

Nemanja SkoricJuly 19th, 2010 - 08:42

You probably think that p1^2 – p2^2 is divisible by 6.

That holds for every p1,p2 > 3, because they are all in form of 6k +- 1. By easy calculation you get what I told.

It is a well known fact actually.

Nemanja SkoricJuly 19th, 2010 - 08:42

You probably think that p1^2 – p2^2 is divisible by 6.

That holds for every p1,p2 > 3, because they are all in form of 6k +- 1. By easy calculation you get what I told.

It is a well known fact actually.

SolJuly 19th, 2010 - 08:48

Nemanja – You’re close but my divisibility observation is stronger than divisibility by 6. Plus, saying that something is “well known” and “by easy calculation” isn’t a proof. Sorry.

Alexander BogomolnyJuly 19th, 2010 - 15:01

Well, p1^2 – p2^2 is divisible by 24 when both primes exceed 3.

In fact p^2 – 1 is divisible by 24 for any such prime.

Of the three numbers

(p – 1), p, (p + 1)

p is odd and prime, implying that both p-1 and p+1 are even, hence one of them is divisible by 4. One of them is also divisible by 3.

SolJuly 19th, 2010 - 20:15

Alex – you got it! Nice simple explanation also.

Alexander BogomolnyJuly 19th, 2010 - 20:38

This is a popular problem. You took it one stem further.

Nemanja SkoricJuly 22nd, 2010 - 06:02

@Sol

It is not proof, I’ve just pointed to it.

But, by skipping that “easy calculation”, I’ve missed the fact that it is divisible by 24.