WA 5 - Nicola Hodges David Walnut MATH 290-002 October 28,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Nicola Hodges David Walnut MATH 290-002 October 28, 2010 Writing Assignment 5 1. Use a form of induction to prove that for all natural numbers n≥2, eithe prime or n has a prime factor p with p ≤ √2. Proof by Complete Induction. Base Case. If n=2, then clearly n is prime and the result holds. Induction Step. Assume it holds for all numbers in the set {2,3,…, n-1}, we will show that it also holds for n. If n is prime, then clearly we are done. Suppose n is not prime, then it has factors other than 1 and itself. Let n=ab, where a and b are natural numbers and 1 < a,b < n. By assumption, the result holds for a and b. It must be true that a ≤ √n or b ≤ √n because if both a >√n and b >√n, then n= ab > (√n) 2 = n which is illogical. So, if both a and b are prime, then n has a prime factor p with p ≤ √n and the results holds. Suppose n has at least one of a or b that is not prime, for clarity assume that it is a. By the induction hypothesis, a has a prime factor p such that p
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

WA 5 - Nicola Hodges David Walnut MATH 290-002 October 28,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online