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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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