Number theory in 21 minutes

> Why is that?

Well for one all prime numbers are odd. And we all know given n (an odd number) n - 1 is even. Providing n > 1 :) (depending on your definition of zero's oddness).

Other than that, it's just a pattern. You could of course replace 24 with any other number that 24 is divisible by (der).

Now all you have to do is prove it inductively -- just to make sure :)

Number theory in 21 minutes

Quote:

Originally Posted by

**zacs7** Well for one all prime numbers are odd.

Wrong. 2 is an even prime number.

Quote:

And we all know given n (an odd number) n - 1 is even.

Right. So?

Greets,

Philip

Number theory in 21 minutes

Quote:

Originally Posted by **Snafuist**

Wrong. 2 is an even prime number.

I think zacs7 forgot the phrase "within the given range" or "greater than 3".

Number theory in 21 minutes

> 2 is an even prime number.

And you specified a range > 3. Last time I checked 2 < 3.

> Right. So?

What do you mean so? It's a key property of the pattern.

Number theory in 21 minutes

Quote:

Originally Posted by **zacs7**

What do you mean so? It's a key property of the pattern.

Right, but honestly I do not see how it proves the property. Could you elaborate?

Number theory in 21 minutes

I never said it proved it :)

Number theory in 21 minutes

Take n where

p = 2*n + 1

p^2 - 1 = (2*n + 1)*(2*n + 1) - 1 = 4*n*n + 2*2*n + 1 - 1

= 4*n*n + 4*n = 4*(n*n + n)

So n*n + n = n*(n+1) must be dividable by 6.

If n is not dividable by 2, then n+1 is. So now we need to proof n*(n+1) is dividable by 3.

If n is not dividable by 3, and n+1 is not dividable by 3, then n+2 must be. Then:

(n+2)*2 = 2*n + 4

must be dividable by 3. And thus

2*n + 4 - 3 = 2*n + 1

must be dividable by 3. However, p = 2*n + 1, and p is a prime number so can't be dividable by 3.

q.e.d.

Number theory in 21 minutes

Right, EVOEx!

Your proof is technically perfect, but it hides the fundamental ideas, so I will present a simpler version:

From school, we remember that p^2 - 1 == (p-1)(p+1)

We want to show that (p-1)(p+1) is dividable by 24, i.e. it's dividable by 2^3 and 3 (the prime factors of 24).

Now have a look at the three consecutive numbers (p-1), p and (p+1). We know that p is a prime > 3, so (p-1) and (p+1) are even. And what's more, if you have two consecutive even numbers, one of them is also dividable by 4. Thus, (p-1)*(p+1) is dividable by 2*4.

Another simple fact about numbers is that of three consecutive integers, one of them is dividable by 3. This is not p (as p is a prime greater than 3), so it must be either (p-1) or (p+1).

Hence, (p-1)*(p+1) is dividable by 2*4 and by 3, i.e. 2*3*4, i.e. 24.

Greets,

Philip

Number theory in 21 minutes

Heh. That really was a lot easier. ;)

Riddle #2: one night in the UAE

This is taken from my exam in "system architecture":

Consider the number abcabc, where a, b and c are arbitrary digits. abcabc is always dividable by 13:

123123 = 13 * 9471

597597 = 13 * 45969

666666 = 13 * 51282

Why is that?

Greets,

Philip

Riddle #2: one night in the UAE

Code:

`a_number_in_the_form_of(abcabc) =`

= c + 10 * b + 100 * a + 1000 * c + 10000 * b + 100000 * a =

= 1001 * c + 10010 * b + 100100 * a =

= 1001 * (c + 10 * b + 100 * a) =

= **13 *** 77 * (c + 10 * b + 100 * a)

Riddle #2: one night in the UAE

Right, anon!

Or to put it in simpler terms:

abcabc = 1001*abc, and 1001 is dividable by 13.

Where do you guys get those byzantine explanations? :P

Greets,

Philip

Riddle #2: one night in the UAE

Quote:

Originally Posted by

**Snafuist** Right, anon!

Or to put it in simpler terms:

abcabc = 1001*abc, and 1001 is dividable by 13.

Where do you guys get those byzantine explanations? :P

Greets,

Philip

Actually, that was the proof I wanted to provide before I read the rest of the posts. However, after typing it in my calculator, I found that "10001" wasn't dividable by 13. I always make those kinds of stupid mistakes....

Riddle #2: one night in the UAE

Quote:

I always make those kinds of stupid mistakes....

Well, you're not alone. My inability to perform mental arithmetic is famous by now...

Your brother,

Philip