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 p12-p22 for most primes p1 and p2? 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!
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July 19th, 2010 - 04:50
I’ve got this so far: http://mathurl.com/34erol5
Will try and work out a proof of it!
July 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!
July 19th, 2010 - 07:21
Can I restrict my observations to the odd primes?
July 19th, 2010 - 07:43
Jonathan – The observation I’m thinking of is not true for all odd primes, just most of them.
July 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.
July 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.
July 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.
July 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.
July 19th, 2010 - 20:15
Alex – you got it! Nice simple explanation also.
July 19th, 2010 - 20:38
This is a popular problem. You took it one stem further.
July 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.