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 December 27th, 2017, 06:08 AM #1 Banned Camp   Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 prime number This is an axiom or a theorem: "There is a number that divide x (that is integer) exist, between 2 to sqrt-x, if x is not a prime number". If this theorem, how it can be proved? Last edited by skipjack; December 27th, 2017 at 11:45 AM. December 27th, 2017, 11:02 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 First, if x is not a prime number, then there exist integers, a and b, such that $\displaystyle ab= x= (\sqrt{x})(\sqrt{x})$. Second, If one of those numbers is greater than $\displaystyle \sqrt{x}$ the other must be less than $\displaystyle \sqrt{x}$. Last edited by Country Boy; December 27th, 2017 at 11:04 AM. December 27th, 2017, 11:04 AM #3 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 It is a theorem based primarily on definitions and the lemma that $1 < \sqrt{m} \implies \sqrt{m} < m.$ $u \text { is a divisor of } w \iff u,\ w \in \mathbb N^+ \text { and } \exists \ v \in \mathbb N^+ \text { such that } u * v = w.$ Do you buy that as a definition? $n \in \mathbb N^+ \text { and } n > 1 \implies n \text { has at least two distinct divisors.}$ That is a theorem easily proved from $n * 1 = n \text { and } n > 1 \implies n \ne 1.$ Now let's define a prime. $p \text { is a prime } \iff p \in \mathbb N^+, \ p > 1,\ \text { and } p \text { has exactly two distinct divisors, namely } 1 \text { and } p.$ OK so far? $\text {Given: } x, y \in \mathbb N+ \text { and } y \text { is a divisor of } x \text { such that } 2 \le y \le \sqrt{x}.$ $\therefore 1 < y < x \implies y \ne 1 \text { and } y \ne x \implies$ $x \text { has at least three distinct divisors } \implies x \text { is not a prime.}$ Last edited by JeffM1; December 27th, 2017 at 11:07 AM. Tags number, prime Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post thinhnghiem Number Theory 15 December 18th, 2016 01:31 PM Dacu Number Theory 3 March 27th, 2014 07:40 AM nukem4111 Number Theory 4 October 7th, 2013 11:29 AM fantom.1040 Algebra 2 June 29th, 2011 03:46 PM

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