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

(In reply to Solution by Federico Kereki)

I've never heard of Newton's formula, so I was wondering how it worked.  I'm sure I could just ask you all, or better yet, ask Google, but I decided to experiment and derive it myself.  Perhaps that English paper I'm procrastinating has influenced my decision.

I tried plugging in a few numbers into excel (1+√2)^n for n=1 to 16 to see what would happen.  The result approached integer, but did not always stay just below it, alternated between being slightly below and slightly above an integer.  This makes sense because (1-√2)^n alternates between negative and positive.

So it appears (1+√2)^n + (1-√2)^n is always an integer.  I guess this makes sense, because if I expand the first part, I get a large integer added to a large multiple of √2.  When I expand the second part, I get a large integer minus that same multiple of √2.  The non-integers cancel.

It seems this should also work with 1 plus the square root of any integer.  It should even work with complex numbers.  A complex number raised to n plus its conjugate raised to the n should be real.

Now this got me thinking...

Posted by Tristan on 2005-04-28 06:44:16