# Euler

#### tank351

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

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#### JeffM1

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

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#### tank351

The number is n $\small\ge$ 13, but I don't know how to prove it. The Ï†(n) = the number of numbers from 1 to n that are relatively prime to n.

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Math Team
7?

1 person

#### tank351

No; if the question were the least number N prime that Ï†(n)$\,\small\ge\,$5 for every n prime $\ge$ N then you'd be right.

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#### v8archie

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

#### skipjack

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

3 people

#### Maschke

The intended problem seems to be to find the least number N such that Ï†(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.

1 person

#### skipjack

Forum Staff
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 Ï†(n).

2 people