Register to reply 
This appears to be in direct violation of Carmichael's theorem. 
Share this thread: 
#1
Mar2014, 03:32 PM

P: 643

In an attempt to prove a statement about the residues of a certain sequence mod ##10^n##, I've derived something which seems to be in direct violation of Carmichael's theorem. Of course, this can't be right, so can someone either explain what bit of my reasoning is wrong or why this isn't in violation of Carmichael's Theorem? First of all, let ##\lambda## be the Carmichael function, and let ##k## be coprime to 2 and 5.
First of all, notice that, by Euler's theorem, ##k^{4\cdot 5^{n1}}\equiv1\pmod{5^n}## and ##k^{2^{n1}}\equiv1\pmod{2^n}##. This makes it clear by induction that ##a\equiv b\pmod{4\cdot 5^{n1}}\rightarrow k^a\equiv k^b\pmod{5^n}## and ##a\equiv b\pmod{2^{n1}}\rightarrow k^a\equiv k^b\pmod{2^n}##. Let ##n\ge2## and ##a\equiv b\pmod{10^n}##. Then, as ##\left.2^{n1},4\cdot 5^{n1}\right10^n##, ##a\equiv b\pmod2^{n1}## and ##a\equiv b\pmod5^{n1}##, so ##k^a\equiv k^b\pmod{2^n}## and ##k^a\equiv k^b\pmod{5^n}##. Therefore ##k^a\equiv k^b\pmod{\mathrm{lcm}\left(2^n,5^n\right)}##, so ##k^a\equiv k^b\pmod{10^n}##. Letting ##a=10^n## and ##b=0##, we get ##k^{10^n}\equiv k^0=1\pmod{10^n}##. As this holds for all ##k## coprime to ##10^n##, this means ##\left.\lambda\left(10^n\right)\right10^n##. (This should be obvious enough; I should be able to provide a proof if necessary.) However, as ##10^n## is not a power of 2, Carmichael's theorem tells us that ##\lambda\left(10^n\right)=\varphi\left(10^n\right)=4\cdot 10^{n1}##, which doesn't divide ##10^n##. Anyone know what's wrong here? 


#2
Mar2014, 06:26 PM

P: 508




#3
Mar2414, 03:56 PM

P: 643

Ah. "A power of an odd prime, twice the power of an odd prime, and for 2 and 4."
*Collides hand with forehead to indicate frustration with self* 


Register to reply 
Related Discussions  
Direct Integration vs. Green's Theorem  Calculus & Beyond Homework  7  
If time travel is possible does it allow for violation of the no clone theorem  Quantum Physics  15  
Violation of Bell's Theorem  Quantum Physics  199  
Noether theorem + CP violation > Energy is not conserved?  Beyond the Standard Model  16  
Carmichael Numbers  Calculus & Beyond Homework  1 