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?!
Thank you
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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 .
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!!!
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.