# Math Help - Primes and Squares

1. ## Primes and Squares

Q: Let p and q be prime numbers. Prove that pq + 1 is a square if and only if p and q are twin primes.

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Pf: p and q are twin primes (i.e. - p = q+2)
Therefore, pq+1 = q(q+2) +1 = q^2 + 2*q +1 = (q+1)^2

(=>)

This is where I'm stuck.

What is a way to show that p and q are not twin primes?
Is it something like p-q > 2?
How would I use something like that to form a contradiction?

2. Originally Posted by dlbsd
Q: Let p and q be prime numbers. Prove that pq + 1 is a square if and only if p and q are twin primes.

(<=)

Pf: p and q are twin primes (i.e. - p = q+2)
Therefore, pq+1 = q(q+2) +1 = q^2 + 2*q +1 = (q+1)^2

(=>)

This is where I'm stuck.
$pq + 1 = x^2$
$pq = (x-1)(x+1)$