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 December 28th, 2018, 06:01 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 07:04 PM.
 December 28th, 2018, 07:00 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,264 Thanks: 1198 Thanks from Maschke and topsquark
 December 28th, 2018, 07:01 AM #3 Senior Member   Joined: May 2016 From: USA Posts: 1,251 Thanks: 519 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 07:04 AM.
 December 28th, 2018, 08:52 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 07:08 PM.
 December 28th, 2018, 09:20 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,561 Thanks: 2562 Math Focus: Mainly analysis and algebra 7? Thanks from topsquark
 December 28th, 2018, 12:59 PM #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 07:03 PM.
 December 28th, 2018, 02:07 PM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,561 Thanks: 2562 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, 07:14 PM #8 Global Moderator   Joined: Dec 2006 Posts: 20,105 Thanks: 1907 The intended problem seems to be to find the least number N such that φ(n)$\,\small\ge\,$5 for every n $\ge$ N. Thanks from Maschke, topsquark and v8archie
December 28th, 2018, 11:11 PM   #9
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Quote:
 Originally Posted by skipjack The intended problem seems to be to find the least number N such that φ(n)$\,\small\ge\,$5 for every n $\ge$ N.
Nice catch. Interesting problem because it's not enough to just look at the table of values and see that 13 seems to work. You have to prove that 13 works; that no number greater than 13 has a totient less than 5. I thought of using Euler's product formula but it's late so maybe someone can supply the proof.

 December 29th, 2018, 12:15 AM #10 Global Moderator   Joined: Dec 2006 Posts: 20,105 Thanks: 1907 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). Thanks from topsquark and v8archie

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