Re: Binomial Theorem for X^n + Y^n

From: Deep K. Deb (deepkdeb_at_yahoo.com)
Date: 08/22/04 

Date: 22 Aug 2004 09:22:13 -0700

analog57@yahoo.com (Russell E. Rierson) wrote in message news:<c410ffa5.0408212104.51a16dfb@posting.google.com>...
> x+y = A
>
> x-y = B
>
> [A+B]/2 = x
>
> [A-B]/2 = y
>
>
> x^2+y^2 = [A^2 + B^2]/2
>
> x^3+y^3 = [A^3 + 3AB^2]/4
>
> x^4+y^4 = [A^4 + 6A^2 B^2 + B^4]/8
>
> x^5+y^5 = [A^5 + 10A^3 B^2 + 5AB^4]/16
>
> x^6+y^6 = [A^6 + 15A^4 B^2 + 15A^2 B^4 + B^6]/32

Very interesting observations. It can further be simplified.
As an example, write x^5 + y^5 = Q5/16
Then x^5 + y^5 = Q5/2^(5-1)
So in general, x^m + y^m = Qm/2^(m-1)
Now, the challenge is to write Qm in more compact form so that it can
be recognized that Qm/(2^m-1) cannot be an m-th power of an integer
without applying FLT.
>
> x^7+y^7 = [A^7 + 21A^5 B^2 + 35A^3 B^4 + 7AB^6]/64
>
> etc...
>
> etc...
>
> etc...

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