Proving sum of root primes is irrational.

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kazafz

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Hey there people, I came across this maths question that I am kind of stuck with. The question is:

If p and q are prime numbers, prove that ?p + ?q is irrational.

I sort of started off the proof by trying to proof by contradiction but got stuck...here is what I've done so far:

Proof by contradiction:

Assume ?p + ?q is rational (i.e Can be written as a fraction m/n where g.c.d (m, n) = 1)

?p + ?q = m/n

p + 2 ?pq + q = m^2 / n^2

Since ?pq is irrational, the sum of it must be irrational too? This is the part im stuck with. I don't know what to do after this.
 
It is well known that if n is a nonsquare positive integer then \(\displaystyle \sqrt{n}\) is irrational.
Assuming that \(\displaystyle p \not= q\) then \(\displaystyle \sqrt{pq}\) is irrational.
 
kazafz said:
Hey there people, I came across this maths question that I am kind of stuck with. The question is:

If p and q are prime numbers, prove that ?p + ?q is irrational.

I sort of started off the proof by trying to proof by contradiction but got stuck...here is what I've done so far:

Proof by contradiction:

Assume ?p + ?q is rational (i.e Can be written as a fraction m/n where g.c.d (m, n) = 1)

?p + ?q = m/n

p + 2 ?pq + q = m^2 / n^2

?pq = [m^2/n^2 - p - q]/2 >>>>> RHS rational but LHS irrational -> Contradiction

Since ?pq is irrational, the sum of it must be irrational too? This is the part im stuck with. I don't know what to do after this.
Click to expand...
 
Thanks Subhotosh, I understand now =D I feel kind of stupid for not knowing that in the first place :lol:
 
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