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April 6th, 2016, 11:46 PM   #1
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prime numbers proof?

Playing with primes numbers I've discovered a seemingly trivial fact
"the smallest non-trivial ( != 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?
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April 7th, 2016, 12:44 AM   #2
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Isn't that a consequence of the Fundamental Theorem of Arithmetic?
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April 8th, 2016, 04:04 PM   #3
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Suppose d, the the smallest divisor of a particular integer is NOT prime.

Then d must have a non-trivial divisor (other than itself or 1).

But this non-trivial 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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