
Math General Math Forum  For general math related discussion and news 
 LinkBack  Thread Tools  Display Modes 
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 sqrtx, 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
The relationships between Prime number and Fibonacci number  thinhnghiem  Number Theory  15  December 18th, 2016 01:31 PM 
A prime number  Dacu  Number Theory  3  March 27th, 2014 07:40 AM 
prime number  nukem4111  Number Theory  4  October 7th, 2013 11:29 AM 
prime number  fantom.1040  Algebra  2  June 29th, 2011 03:46 PM 