Question

In base 6, the first 7 digits of any numbers x and y obey this... Why? ;)

Answer 1

\[\ (x1350213 + y4205344)^n = x^n1350213 + y^n4205344\]

Answer 2

If I get the answer will you give me Owl bucks?

Answer 3

M A Y B E

Answer 4

Liar e.e

Answer 5

I've gotta go to sleep, I'll be back to check up on this later. ;P

Answer 6

will try

Answer 7

Notice that in base-6, we have \(4205344 = 6^7-1350213 +1\)

\[\begin{align}\ [x1350213 + y4205344]^n
&= [6^7(x+y) +1350213 +4205344 ]^n\\~\\
&= [6^7(x+y) +6^7+1 ]^n\\~\\
&=6^7A + 1
\end{align}\]

For the given number format, the right most 7 digit string is \(0000001\) no matter what \(x\) and \(y\) are.

Answer 8

Wait a sec that doesn't seem right, here's a counter example picked at random!
\[(3*1350213 + 5*4205344)^2 = 0330124521\] all in base 6 still.

I think you're onto some good ideas though.

Answer 9

Hmm whoops I put the answer in base 10 on the right there.

I need a better way of going between base 6 and not. Something doesn't seem right overall, maybe I'm mistaken here.

Answer 10

Alright I'll give my solution since I worked it out after confusing myself and want to just get it out there before I forget.

http://www.wolframalpha.com/input/?i=%5B(3%2B1*6%2B2*6%5E2%2B0*6%5E3%2B5*6%5E4%2B3*6%5E5%2B1*6%5E6)%5E2%5D+base+6

This shows that for the first several digits in base 6,
\[(1350213)^2 = 1350213\]
I'll go ahead and name it based on the digit it ends in, \(3\) as:
\[i_3 = 1350213\]
So it obeys:\[i_3^2=i_3\]
Similarly the other number
\[i_4=4205344\]obeys\[i_4^2=i_4\]
Now the nail in the coffin is:
\[i_3i_4=0\]
So now it automatically will collapse this sucker down, all the cross terms go away!\[\ (x1350213 + y4205344)^n = x^n1350213 + y^n4205344\]

Answer 11

Jesus man.

Answer 12

Cool right? If only we had numbers that infinitely extended to the left, then it would apply not just to the first 6 or so digits, it'd apply to all digits...!

Answer 13

Ok, so basically I cooked this question up after playing around with 6-adic numbers, these are approximations to some of those numbers. :P