December 28th, 2018, 05:01 AM  #1 
Newbie Joined: Dec 2018 From: israel Posts: 5 Thanks: 0  Euler
The least number n that φ(n) $\small\ge$ 5?
Last edited by skipjack; December 28th, 2018 at 06:04 PM. 
December 28th, 2018, 06:00 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389  
December 28th, 2018, 06:01 AM  #3 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 
What is the definition of the totient function? What numbers did you explore? Did you see any patterns?
Last edited by JeffM1; December 28th, 2018 at 06:04 AM. 
December 28th, 2018, 07:52 AM  #4 
Newbie Joined: Dec 2018 From: israel Posts: 5 Thanks: 0 
The number is n $\small\ge$ 13, but I don't know how to prove it. The φ(n) = the number of numbers from 1 to n that are relatively prime to n.
Last edited by skipjack; December 28th, 2018 at 06:08 PM. 
December 28th, 2018, 08:20 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
7?

December 28th, 2018, 11:59 AM  #6 
Newbie Joined: Dec 2018 From: israel Posts: 5 Thanks: 0 
No; if the question were the least number N prime that φ(n)$\,\small\ge\,$5 for every n prime $\ge$ N then you'd be right.
Last edited by skipjack; December 28th, 2018 at 06:03 PM. 
December 28th, 2018, 01:07 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 
I don't see why you want to put so many conditions in there. But whatever.

December 28th, 2018, 06:14 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,919 Thanks: 2202 
The intended problem seems to be to find the least number N such that φ(n)$\,\small\ge\,$5 for every n $\ge$ N.

December 28th, 2018, 10:11 PM  #9  
Senior Member Joined: Aug 2012 Posts: 2,355 Thanks: 736  Quote:
 
December 28th, 2018, 11:15 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,919 Thanks: 2202 
It's probably easier to use a number considerably greater than 13, then verify the result for lower numbers by reference to a list of values of φ(n).


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