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December 28th, 2018, 05:01 AM   #1
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Euler

The least number n that φ(n) $\small\ge$ 5?

Last edited by skipjack; December 28th, 2018 at 06:04 PM.
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December 28th, 2018, 06:00 AM   #2
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Euler's Totient Function Values For n = 1 to 500, with Divisor Lists
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December 28th, 2018, 06:01 AM   #3
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What is the definition of the totient function? What numbers did you explore? Did you see any patterns?

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December 28th, 2018, 07:52 AM   #4
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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.
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December 28th, 2018, 08:20 AM   #5
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December 28th, 2018, 11:59 AM   #6
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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.
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December 28th, 2018, 01:07 PM   #7
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I don't see why you want to put so many conditions in there. But whatever.
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December 28th, 2018, 06:14 PM   #8
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The intended problem seems to be to find the least number N such that φ(n)$\,\small\ge\,$5 for every n $\ge$ N.
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December 28th, 2018, 10:11 PM   #9
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Quote:
Originally Posted by skipjack View Post
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.
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December 28th, 2018, 11:15 PM   #10
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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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