Proof

Question 1

Proof of infinite primes.

Answer 1

• Assume there is a finite number of primes.
• List them P1,P2 … Pn as highest prime.
• To find new prime: P1 x P2 x … x Pn +1
• Case 1: N is a new prime number.
• Case 2: N is not prime but product has new prime factors not listed already.
• In all cases we have a contradiction as we have new primes.
• Hence there must be infinite primes.

Question 2

Proof that sqrt(2) is an irrational number.

Answer 2

• Assume sqrt(2) is rational.
• Therefore we could say sqrt(2) = a/b
• 2 = a^2/b^2, 2b^2 = a^2
• In the prime factorisation of a^2, 2 occurs an even number of times but in the prime factorisation of b^2, 2 occurs an odd number of times.
• But as 2b^2 = a^2 they should have the same prime factorisation.
• Contradiction, hence sqrt(2) must be irrational.