Nicola Hodges
David Walnut
MATH 290002
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,…, n1}, 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
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 Fall '08
 Alligood,K
 Advanced Math, Natural Numbers, Mathematical Induction, Natural number, Peano axioms, Nicola Hodges David Walnut

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