Wednesday, 12 August 2015

number theory - Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if p1, p2 and p3 are distinct primes then p1+p2 and p1+p2+p3 are irrational, but the only proof I can find that p1+p2+...+pn is irrational for distinct primes p1, p2, ... , pn requires we consider finite field extensions of Q.



Is there an elementary proof that p1+p2+...+pn is irrational exist?



(By elementary, I mean only using arithmetic and the fact that m is irrational if m is not a square number.)



The cases n=1, n=2, n=3 can be found at in the MSE question sum of square root of primes 2 and I am hoping for a similar proof for larger n.

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