Question

For N ∈ Z +, consider the following statement: P(N) : ∀x ∈ Z, N | x 2 =⇒ N | x. Note that we have already proved P(2) and P(3) – these were Propositions 6.5 and 6.6 in the course text (and covered in class). a) Explain why our proof of the irrationality of √ 2 (see Theorem 6.8 in the course text) used P(2) in a crucial way. b) Using the fact that P(3) is true,

Answer 1

The proof of the irrationality of √2 used P(2) to establish the contradiction needed to prove its irrationality. Using P(3), we can show that if n^2 divides x^2, then n divides x.

Step-by-step explanation:

The proof of the irrationality of √2 (Theorem 6.8 in the course text) used P(2) in a crucial way because it showed that if √2 was rational, then there exists a fraction whose square is equal to 2. This contradiction was used to prove that √2 is irrational. By assuming P(2) to be true, the proof was able to establish the irrationality of √2.

To use the fact that P(3) is true, we can show that if n is a positive integer and x is an integer such that n^2 divides x^2, then n divides x. This can be proven by considering the prime factorization of n and x and using the fact that the exponents of the prime factors must be greater in n^2 than in x^2, which implies that n divides x.