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The Magic Cafe Forum Index » » Magical equations » » Prime number peculiarity (2 Likes) Printer Friendly Version

SteveFowkes
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It is a fact that any given prime number when squared and 1 taken away is EXACTLY divisible by 24. i.e (n^2-1)/24 = Integer (where n is a prime number).

The question is why does this work?

I genuinly don't know the answer but I'm hoping someone can tell me.

Hoping for a solution,
Steve
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TomasB
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I think I can reason my way through this. Write it as (n - 1)(n + 1)=N * 2 * 2 * 2 * 3 where n is a prime bigger than 3 and N is an integer.

We know that n is odd. That means that either (n - 1) or (n + 1) is dividable by 4 while the other is dividable by 2. That accounts for the three 2-factors in 24.

Of any three concecutive integers exactly one is dividable by 3. Since n wouldn't be prime if it was dividable by 3 it has to be either (n - 1) or (n + 1) that is.

/Tomas
SteveFowkes
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Tomas,

It took me a while but by putting some numbers into a calculator, the penny dropped that (n-1)(n+1) was actually n^2-1 and NOT n^2. Duh!

Working with your solution, and correct me if go astray, I get the following.

Ignore n as it won't divide by anything (prime)
n-1 and n+1 divide by 2
n-1 or n+1 also divides by 4
n-1 or n+1 also divides by 3
So by multiplying factors together (^2), we get either

(2*3*4)(2) = 48
(2*3)(2*4) = 48

Obviously they'll divide by 24 but it looks like they divide by 48 as well. I've tried a bit of manual number crunching and this seems to work. Am I correct?

This is a problem that has been bugging me for a number of years now. It was set by my Mathematics lecturer whilst doing an Aeronautical Engineering degree and he never told us the solution.

Many thanks for your help. It really is most appreciated.

Steve
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SteveFowkes
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Sorry, a bit more number crunching found that it DOESN'T always divide by 48. Where did I go wrong?
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rgranville
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As you noted, n-1 AND n+1 are divisible by 2. EITHER n-1 OR n+1 is divisible by 4. But that 4 "includes" the 2. You're multiplying that 2 in twice.

Let me give an example. Take the prime number 11. n-1 = 10. n+1 = 12. Obviously, both 10 and 12 are divisible by 2. Obviously 12 is divisible by 4, as well as by 3. But n-1 * n+1 = 10 * 12 = 120 is NOT divisible by (2*3*4)*2 = 48. Why?

10 = 2 * 5 (prime factorization). 12 = 2 * 2 * 3, which equals 4 * 3, NOT
2 * 4 * 3. 12 is divisible by 2 and by 4 but NOT 2 * 4. The 4 "includes" the 2.

Going back to the general, either n-1 or n+1 is divisible by 2, but not by 4. The other is divisible by 4 - which implies divisibility by 2 but NOT that it is divisible by 2 * 4 = 8. Either n-1 or n+1 is divisible by 3. So n-1 * n+1 has to be divisible by 2 * 3 * 4 = 24.

As an aside, note that if n is any odd number that isn't a multiple of 3, n^2 - 1 will be divisible by 24

:pepper:
SteveFowkes
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I can see the mistake now. Thanks for that.

I'm a better engineer than a mathematician - honest!

Mezomagic,

I can feel for you having to give a presentation that even the audience doesn't particularly want to hear. I've had to do a few myself for various subjects and offer these tips, of which I've personally used.

Hidden content. Try to put as many song titles within your presentation as possible. If nobody notices, it's a little fun for yourself. The chances are however, that a few or more will notice and actually listen out for them all. It's better to put a couple of abstract ones in early that don't generally fit with the presentation to 'give the game away' and get them listening. Other subjects work but song titles give more scope.

Magic props. I tend to pick up a fire book during the presentation as if to read a quote and completely ignore the flames. Don't mention them at all. Also, a break-away wand. Point to something with it and just let it flop. Use a TT to generate a hanky to blow your nose and get rid of it again also works if done in a completely blase way. Various other minor props used but not mentioned, again will make them wonder what's next.

Comedy. Fiscal policy isn't exactly the top of any stand-up comics short list but the occasional pun or two may be a bit of light relief. Also, the golden rule of presentations is never turn your back on your audience. But, a sign on your back saying 'Are you bored yet' revealed about half way through could produce a reaction.

Aside from that, if you scour the various topics within the Café, there's plenty of material proving 2=1 and the like.

If all else fails, you could gradually remove your clothes during the presentation to eventually reveal bra, stocking and suspenders (I'm assuming you're male). I've not done this but was part of an audience during which someone did. The subject was particularly boring but I still remember it now - it was 6 years ago!

It really does depend on who you're presenting to but I wish you well.

Regards,
Steve
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Steve
Scott Cram
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I know I’m reaching back a bit for this thread, but I was reminded of it when Numberphile posted their latest video: https://www.youtube.com/watch?v=ZMkIiFs35HQ
Larry Barnowsky
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In chapter 19 of my book Magica Analytica II, there is a card effect called The Prime Directive. It uses the fact that a prime squared minus 1 is always divisible by 3 and in fact 24. I include a proof of that in the chapter.

Larry (my password Pi)
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