(Worried)

Determine all integers n for which Euler's function (phi) = n/2?

Can anyone please help me, I'm really not sure what to do?!(Crying)

Thank you

xx

- August 18th 2010, 03:38 AMHorseStar2Can any one please help with this Euler's function question ?
(Worried)

Determine all integers n for which Euler's function (phi) = n/2?

Can anyone please help me, I'm really not sure what to do?!(Crying)

Thank you

xx - August 18th 2010, 06:52 AMsimplependulum
We have this famous result where .

Then we are going to show . The key to the problem is using inequality :

We know the smallest prime is , thus or

Therefore , where denotes the number of primes involving in the product , since , we must have and the only one prime is . We find that is the only prime divisor of so it is a power of namely . - August 18th 2010, 07:12 AMthorin
Hi,

if you don't know the formula used by simplependulum, you can also write , with , and use when - August 19th 2010, 10:23 PMHorseStar2
Hello thorin, thank you very much for your help, I wondered if you could please solve the question using the method you suggested too please as I'm not sure how to do it?

Also, (sorry to ask soo much) but I am also trying to solve..

Determine all integers n for which Euler's (phi) function = n/6

and I cannot do it, can you please explain ho to do this too?

THANK YOU SO MUCH!!! (Itwasntme) - August 19th 2010, 10:27 PMHorseStar2
- August 19th 2010, 10:59 PMundefined
I believe we can prove there are no such integers.

Suppose we write as its prime decomposition where as usual . Consider the product which must equal to satisfy the given condition. When is written as a fraction in lowest terms, the denominator must be divisible by . (Details left for you to fill in.) So n cannot be divisible by a prime greater than 3. There are a trivially small number of cases to test to complete an exhaustive search. - August 19th 2010, 11:39 PMHorseStar2
- August 19th 2010, 11:44 PMundefined
- August 19th 2010, 11:47 PMHorseStar2
- August 19th 2010, 11:49 PMundefined