April 6th, 2016, 11:46 PM  #1 
Member Joined: Mar 2016 From: Nepal Posts: 37 Thanks: 4  prime numbers proof?
Playing with primes numbers I've discovered a seemingly trivial fact "the smallest nontrivial ( != 1) divisor of any positive integer is a prime". Does the proof of this exist? Sure it should unless my conclusion is wrong. Anybody else already know about this idea? 
April 7th, 2016, 12:44 AM  #2 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics 
Isn't that a consequence of the Fundamental Theorem of Arithmetic?

April 8th, 2016, 04:04 PM  #3 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 393 Thanks: 71 
Suppose d, the the smallest divisor of a particular integer is NOT prime. Then d must have a nontrivial divisor (other than itself or 1). But this nontrivial divisor would have to be smaller than d. Therefore d is NOT the smallest divisor of that integer. This implies that the original supposition is false, and thus the smallest divisor cannot be composite, and so it must be prime. 

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numbers, prime, proof 
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