Prove that p + q is an irrational, where p , q are primes.

Let us assume that p + q is a rational number. Then, there exist co-prime positive integers a and b such that, p + q = a b ; b ≠ 0 ⇒ a b - p = q Squaring both sides, ⇒ a b - p 2 = ( q ) 2 ⇒ a 2 b 2 - 2 a b p + p = q ⇒ a 2 b 2 + p - q = 2 a b p ⇒ a 2 + p - q b 2 b 2 = 2 a b p ⇒ a 2 - p - q b 2 2 a b = p Here, L.H.S. is a rational number while p is irrational because p is prime. Hence, our assumption is wrong. Thus, p + q is an irrational number.

Prove that p + q is an irrational, where p , q are primes.

Let us assume that p + q is a rational number. Then, there exist co-prime positive integers a and b such that, p + q = a b ; b ≠ 0 ⇒ a b - p = q Squaring both sides, ⇒ a b - p 2 = ( q ) 2 ⇒ a 2 b 2 - 2 a b p + p = q ⇒ a 2 b 2 + p - q = 2 a b p ⇒ a 2 + p - q b 2 b 2 = 2 a b p ⇒ a 2 - p - q b 2 2 a b = p Here, L.H.S. is a rational number while p is irrational because p is prime. Hence, our assumption is wrong. Thus, p + q is an irrational number.