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Math Help - Primes and Squares

  1. #1
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    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?
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  2. #2
    Super Member
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    Quote Originally Posted by dlbsd View Post
    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
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