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.
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.