The discussion centers on proving that for any odd composite number \( n \), all of its prime factors are at most \( \frac{n}{3} \). Key points include the application of the fundamental theorem of arithmetic, which states that every integer can be expressed as a product of primes. Participants emphasize the importance of recognizing that the smallest prime factor of an odd composite number must be 5, leading to the conclusion that \( n \) is divisible by 5. Additionally, Euclid's lemma is referenced as a crucial concept for connecting \( n \) with its prime factors.
PREREQUISITESMathematics students, educators, and anyone interested in number theory, particularly those studying properties of composite numbers and prime factorization.
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?