What is the 1000th digit to the right of the decimal point in the decimal representation of (1+√2)^3000?

This problem can be solved by algebra alone, without the need for computers or calculators

I started looking at powers of (1+¡Ì2)

1+¡Ì2
3+2¡Ì2
7+5¡Ì2
17+12¡Ì2
41+29¡Ì2

these numbers are of the form A + B¡Ì2
where A(n) = A(n-1) + 2B(n-1)
and B(n) = A(n-1) + B(n-1)

Then I looked at the decimal for 41+29¡Ì2 ¡Ö82.01219 pretty close to 41*2.  In fact it gives 41/29 as a pretty good approximation for ¡Ì2.

Looking at the next few terms

99 + 70¡Ì2 ¡Ö 197.99495
239 + 169¡Ì2 ¡Ö478.00209

They seem to approach integers pretty quickly from above and below.  I'm not sure if they approach quickly enough, but if they do, the even terms are slightly below integers.

I'm tempted to guess the 1000th decimal place of the 3000th power is a 9.

 

Posted by Jer on 2005-04-26 19:10:52