The discussion revolves around proving that for any odd composite number ##n##, all of its prime factors are at most ##\frac{n}{3}##. Participants are exploring the implications of prime factorization and the properties of composite numbers.
There is an ongoing exploration of different approaches to the problem, with some participants suggesting connections between the properties of primes and composite numbers. While some hints and ideas have been shared, there is no explicit consensus on a method or solution yet.
Participants are navigating assumptions about the nature of odd composite numbers and the use of established theorems in their reasoning. There is a mention of potential constraints regarding the professor's instructions on what can be assumed or used in the proof.
Are you allowed to use the prime decomposition of integers?GlassBones said:Homework Statement
Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ##
Homework Equations
Some theorems might help?
Any ##n>1## must have a prime factor
if n is composite then there is a prime ##p<√n## such that ##p|n##
The Attempt at a Solution
I'm at a standstill, not sure how to start off this question? Can anyone prove hints to get me going
The fundamental theorem of arithmetic says, that every integer can be written as a product of primes. If you are allowed to use this, and proofs can easily be found, e.g. on Wikipedia, then you only have to write this in a formula and think about what the smallest prime in there is.GlassBones said:I'm pretty sure I can (prof didn't talk about it but I'm assuming we should know about it).
With prime decomposition, I'm not sure how to generate it. Can we just assume there is a prime decomposition because n is composite?