Euler

Dec 2018
5
0
israel
The least number n that φ(n) $\small\ge$ 5?
 
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May 2016
1,310
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USA
What is the definition of the totient function? What numbers did you explore? Did you see any patterns?
 
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Dec 2018
5
0
israel
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.
 
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Dec 2018
5
0
israel
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.
 
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v8archie

Math Team
Dec 2013
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I don't see why you want to put so many conditions in there. But whatever.
 

skipjack

Forum Staff
Dec 2006
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2,470
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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Aug 2012
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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.
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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skipjack

Forum Staff
Dec 2006
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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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