Thread: I have problem about primes.(Help me please!)

If p>=q>=5 and p and q are both primes, prove that 24|(p^2-q^2).



Thank.

Originally Posted by konna

If p>=q>=5 and p and q are both primes, prove that 24|(p^2-q^2).



Thank.

What have you tried yourself? Of course, since p and q are both primes larger than 3, they are both odd and so both p-q and p+q are even- their product is, at least, divisible by 4. Now see if you can find another factor of 2. If p= 2m+1 and q= 2n+1, p+q= 2m+2n+ 2= 2(m+n+1) and p- q= 2m- 2n= 2(m-n). Consider what happens to m+n+1 and m-n if m and n are both even, both odd, or one even and the other odd. Notice that so far we have only required that p and q be odd, not that they be prime.

Once you have done that the only thing remaining is to show that must be a multiple of 3 and that means showing that either p-q or p+q is a multiple of 3.

Originally Posted by konna

If p>=q>=5 and p and q are both primes, prove that 24|(p^2-q^2).



Thank.

, and as:

1) not a multiple of 3 ;

2) either or is divisible by 4, and the other one by 2.

The result follows at once.

Tonio

I used to try by myself , but i not sure my process.

Thank so much ..

Hello, konna!

I don't know if this qualifies as a proof.

If and and are both primes,

. . prove that: .

Any prime greater than or equal to 5 is of the form: .
. . for some integer


Let: .

Then: .

Subtract: .

. . . . . . .
. . . . . . .


We see that is divisible by 12.

We must show that either or is even.


If and has the same parity (both even or both odd),
. . then is even.


If and have opposite parity (one even, one odd),
. . then is odd.
And . is odd.
. . Then: . is even.


Therefore, is divisible by 24.