Prime^2 -1

[André Grisell]

p is a prime >3. Show that p^2-1 is divisible by 24.

Let's check a few:

5^2-1 = 25-1 = 24. Correct!
7^2-1 = 49-1 = 48 = 2x24. Correct!
11^2-1 = 121-1 = 120 = 5x24. Correct!
13^2-1 = 169-1 = 168 = 7x24. Correct!
17^2-1 = 289-1 = 288 = 12x24. Correct!

And so on. It seems to work. Now please show that this works for all primes >3.

 

[MarkFL]

Consider that:

[MATH]p^2-1=(p+1)(p-1)[/MATH]
Primes above 3 are all odd, and so the two factors must both be even, and every other even number is divisible by 4, and so the expression is divisible by 8. Since \(p\) cannot be a multiple of 3, one of the two factors must be, and so we are guaranteed the expression is divisible by 24.

 

[Steven G]

p is a prime >3. Show that p^2-1 is divisible by 24.

Let's check a few:

5^2-1 = 25-1 = 24. Correct!
7^2-1 = 49-1 = 48 = 2x24. Correct!
11^2-1 = 121-1 = 120 = 5x24. Correct!
13^2-1 = 169-1 = 168 = 7x24. Correct!
17^2-1 = 289-1 = 288 = 12x24. Correct!

And so on. It seems to work. Now please show that this works for all primes >3.

Hi, I am sorry but whether you ask nicely or not you can not just ask us to do your problems for you. This is a math help forum where we (usually) expect the student to solve their own problem with help from the forum's helpers. That means that you need to show us what you have tried and inform us where you are stuck. In this way we know what kind of help you need. Please post back with the work you understand with this problem including what you learned from MarkFL's post so we know if you understand the work.

 

[André Grisell]

Hi, I am sorry but whether you ask nicely or not you can not just ask us to do your problems for you. This is a math help forum where we (usually) expect the student to solve their own problem with help from the forum's helpers. That means that you need to show us what you have tried and inform us where you are stuck. In this way we know what kind of help you need. Please post back with the work you understand with this problem including what you learned from MarkFL's post so we know if you understand the work.

Okay. Thank you. I thought that I had the solution myself. Just wanted to make sure it was correct. And I got the reply from MarkFL that was consistent with my assumption.

Thank you.

 

[HallsofIvy]

Okay. Thank you. I thought that I had the solution myself. Just wanted to make sure it was correct. And I got the reply from MarkFL that was consistent with my assumption.

Thank you.

Do you really expect anyone to believe that you had the solution yourself? If you do then the reasonable thing to do woud be to post it and ask if it is correct.