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 tank351 December 28th, 2018 05:01 AM

Euler

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

 romsek December 28th, 2018 06:00 AM

Euler's Totient Function Values For n = 1 to 500, with Divisor Lists

 JeffM1 December 28th, 2018 06:01 AM

What is the definition of the totient function? What numbers did you explore? Did you see any patterns?

 tank351 December 28th, 2018 07:52 AM

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.

 v8archie December 28th, 2018 08:20 AM

7?

 tank351 December 28th, 2018 11:59 AM

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.

 v8archie December 28th, 2018 01:07 PM

I don't see why you want to put so many conditions in there. But whatever.

 skipjack December 28th, 2018 06:14 PM

The intended problem seems to be to find the least number N such that φ(n)\$\,\small\ge\,\$5 for every n \$\ge\$ N.

 Maschke December 28th, 2018 10:11 PM

Quote:
 Originally Posted by skipjack (Post 603764) 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.

 skipjack December 28th, 2018 11:15 PM

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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