Algebraic Proof about Product of Twin Primes

Date: 04/23/2008 at 15:18:02
From: Nicole
Subject: Product of twin primes + 1 = perfect square, proof

I have a question regarding a prime proof.  Say for instance 1 is 
added to a product of twin primes.  How can I prove that a perfect 
square will always be obtained?  Thank you.

Date: 04/23/2008 at 19:57:33
From: Doctor Achilles
Subject: Re: Product of twin primes + 1 = perfect square, proof

Hi Nicole,

Thanks for writing to Dr. Math.

Twin primes are two consecutive odd integers that are both prime.  If
we call the first integer in the pair "p" then the second is p + 2.

Therefore the product is:

  p(p + 2)

or

  p^2 + 2p

If we add 1 to the product we have:

  p^2 + 2p + 1

Is that always a perfect square?

If you're stuck, try a few examples.  What is the square root of
11*13+1?  What about 5*7+1 or 3*5+1?

Hope this helps.  If you have other questions or you're still stuck
and you'd like to talk about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/