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Question

Question 14
Prove that p+q is irrational, where p and q are primes.


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Solution

Let us suppose that p+q is rational.
Again, let p+q=a, where a is rational.
Therefore, q=ap
On squaring both sides, we get
q= a2+p2ap [ (ab)2=a2+b22ab]
Therefore, p=a2+pq2a.

This contradicts our assumption, as the right-hand side is a rational number whereas p is irrational.
Hence, p+q is irrational.


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