From: lerma@math.nwu.edu (Miguel A. Lerma)
Subject: Re: Almost Integers
Date: 28 Jul 2000 15:45:42 GMT
Newsgroups: sci.math
Summary: [missing]

Oscar Lanzi III (ol3@webtv.net) wrote:
: Let a and b be positive numbers satisfying a^2 - b = +/- 1.  Raise
: a+sqrt(b) to the nth power, and divide by 2.  As n increases without
: bound, the results come arbitrarily close to being integers.  Anyone
: familiar with quadratic surds knows why.

More generally a similar property holds for Pisot-Vijayaraghavan 
numbers, i.e., real algebraic integers greater than 1 such that 
their conjugates are inside the unit complex circle. If x is a 
P-V number and x1,...,xk are its conjugates, then 

        x^n + x1^n + ... + xk^n

is an integer for any positive exponent n. Since x1^n + ... + xk^n
tends exponentially to zero as n goes to infinity, x^n comes very 
close to be an integer as n increases.

The simplest example of P-V number is the golden ratio phi =
(1+sqrt(5))/2 = 1.618033989, whose conjugate is (1-sqrt(5))/2 =
-.6180339890. We notice for instance that

((1+sqrt(5))/2)^100 = 792070839848372253126.999999999999999999998737...

is in fact very close to an integer.

Another example of P-V number is the real root of x^3-x-1=0, which 
is 1/6*(108+12*69^(1/2))^(1/3)+2/(108+12*69^(1/2))^(1/3).


Miguel A. Lerma