# Thread: 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.

(<=)

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.
Pf: (By Contradiction?)

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.
Pf: (By Contradiction?)

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?
$pq + 1 = x^2$
$pq = (x-1)(x+1)$

using unique prime factorization result, we can get the required proof as well

### Prove that pq 1 is a square if and only if p and q are twin primes.

Click on a term to search for related topics.