Prove that for any distinct primes p and q, the ratio p q is irrational.

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Prove that for any distinct primes   p and   q, the ratio             p              q       is irrational.

trantegisis · Accepted Answer

Suppose that you can write             p      q       Where   p and   q are primes and   a,  b are rational numbers and that       a    b   cannot be reduced.Then       p    q    =            a      2              b      2      (1)       p    q    ∗            b      2        =            a      2      Then a is a multiple of pq.(2)   a  =  k  ∗      p    q  Insert (2) in (1):       p    q    ∗            b      2        =      (    k    ∗          p      q              )      2            p    q    ∗            b      2        =            k      2        ∗                  p        2                    q        2            Divide by       p    q  :            b      2        =            k      2        ∗          p      q      We see that both   b and   a have common factors, but due to our initial statement that   b and   p have no common factors,             p      q       is not rrational.

Prove that for any distinct primes p and

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