Euler's theorem - conjecture (is it true?)

darij grinberg

The author of this posting is : Krishna Kothoor
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Euler's theorem: If N is any natural number and $\phi\left(N\right)$ is the number of intergers in the sequence 1,2,3,....N-1 which are relatively prime to N then $a^{\phi\left(N\right)}-1$ is divisible by N where a is any integer which is relatively prime to N.

My conjecture:
For any prime number p there exists no k less than $\phi\left(p\right)$ such that $2^k-1$ is divisible by p.

I considered several examples of powers of 2 but only upto a limit because after that calculations become pretty laborius, but i am very confident about this conjecture being true.
Does any one agree with this?
Any comments or validations are welcomed.

**Posted:** Fri Dec 10, 2004 7:58 pm

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zscool · Riemann Hypothesis Offline Joined: 22 Jun 2003 Posts: 482

um 7 = 2^3 - 1? unless im mistaking your question

**Posted:** Fri Dec 10, 2004 10:46 pm

luther_driggers · New Member Offline Joined: 10 Dec 2004 Posts: 19

i do belive zscool is right. r u sure u stated the idea correctly?

**Posted:** Sat Dec 11, 2004 11:59 pm

Valentin Vornicu

Any $k$ less than $\varphi (p)$ means any $k<p-1$ . Obivously that statement is not true, as the order of 2 modulo $p$ is not always $p-1$ .

Let's try thinking about this if $p$ is not a prime number (then $\varphi(p)<p-1$ ... altough I doubt it there too.

**Posted:** Sun Dec 12, 2004 7:23 am

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luther_driggers · New Member Offline Joined: 10 Dec 2004 Posts: 19

seems that doesn't work either.

$n=21$

$2^6 \equiv_{21} 1 \Rightarrow n|2^6-1$

hehe... it's true for even numbers Mr. Green

**Posted:** Sun Dec 12, 2004 11:47 pm

mlee1

the "conjecture" says ord(2) = \varphi (p) which I don't believe is true in general.
It seems like it for some examples just beause 2 is small.
In fact, I seem to remember having encountered that claim as a joke, because 2 seems to be the first example everybody tries when trying to find a generator for the unit group.

**Posted:** Mon Jan 10, 2005 4:06 pm