From - Fri Nov 28 11:50:45 1997
From: jhnieto@luz.ve
Subject: Re: Challenge Problem Correction and Clarification
Message-ID: <880335931.30735@dejanews.com>

>  Can you prove that  if all combinations of the following
>  (+-Sqrt(x_1)+-Sqrt(x_2)+...+-Sqrt(x_n)) are MULTIPLIED together.
>  where +- is + or -, then the result is the square of an integer
>  where x_1,x_2,..., x_n are all integers?

Consider the polynomial

P(z_1,z_2,...,z_n) = product of the 2^n sums +-z_1+-z_2+-...+-z_n

If n>1 it's easily seen that P is the square of the polynomial

Q(z_1,z_2,...,z_n) = product of the 2^(n-1) sums z_1+-z_2+-...+-z_n

Now  Q(-z_1,z_2,...,z_n) = Q(z_1,z_2,...,z_n)  which implies that z_1 has
even exponent in each monomial of Q, and the same is true for z_2,...,z_n.
Since Q has integer coefficients it follows that if x_1,x_2,...,x_n are
non negative integers then Q(sqrt(x_1),sqrt(x_2),...,sqrt(x_n)) is an
integer, hence P(sqrt(x_1),sqrt(x_2),...,sqrt(x_n)) is a square.

José H. Nieto