Re: A prove about Euler number
I've looked around a little. I'm not sure about part (b) of your
question, since it looks a little strange to me. But anyway.
As for part (a), I'd suggest you look up the topic of 'primitive
roots' in any good number theory book. I have a proof in front of me
of the non-existence of primitive roots for a certain type of moduli,
and the proof uses something very similar to your number 'U'.
(If you want specifics, the book is that of Niven, Zuckerman &
Montgomery, & I'm looking at the proof of Thm 2.41. )
Hope this is of help, sorry I can't say much about part (b)!
--- In firstname.lastname@example.org, "Bill" <iampoliceman@y...> wrote:
> let n = (P1^k1)(P2^k2)....(Pl^kl) where the Pi's are distinct primes
> and where each Ki >= 1,and let U = lcm(O(P1^k1),...,O(Pl^kl)), where
> O is the Euler number.
> a) show that if gcd(a,n) = 1, then a^U = 1 (mod n)
> b) show that for any a we have a^(U+m) = a^m (mod n), where m = max
> any help is greatly appreicated!!! thanks