higher powers of(2+SQR3)differ negligibly from whole nos.

[Don McDonald]

example.
V3 = sqrt3.
(2+V3)(2-V3) = (4-3)   = 1.
2-V3 ~ 0,  approximately zero.

square both sides..
(7+4V3)(7-4V3) = 1

7-4V3 is closer to zero.
7+4V3 is closer to 2*7.

therefore, higher powers of (2+V3)^n ever approach
larger even integers.
= x ^ -n, where x= (2-V3) is small << .5.
and sqrt3 is approximated by (rational) convergents.

recurrence relation.
(a+bV3)(2+V3)
=  2a+3b  +  (a+2b)V3.
-> 4a+6b.

don.mcdonald
21.01.03    22.01.03    19:12 nzdt*.
============
my file > .Calc.Profile.eisintegsq.Seqfan.ogerardMod.2+V3


?2+SQR3
1) 3.73205081  Optn Y
: disp Roots x^(1/y),& powers x^y.
			*******

1        3.73205081     3.73205081
2        1.93185165     13.9282032
3        1.55113352     51.9807621
                         ***
4        1.38991066     193.994845
5        1.3013362     723.998619
6        1.24544511     2701.99963
7        1.20699867     10083.9999
8        1.17894472     37634
9        1.15757657     140452
10        1.14076124     524174
11        1.12718508     1956244
index y |  x^(1/y)  |    x^y
                         ***
                         
 HIGHER POWERS OF 2+SQR3 DIFFER NEGLIGIBLY FROM WHOLE NOS.

To: primeness at borve.demon.co.uk
Subject: Re: log(m)'s close to integers, various log bases

In message <urvRMmA0bUL+Ew+2 at borve.demon.co.uk> you write:

> In message <LW9ntHj034n at paradise.net.nz>, Don McDonald
> <parabola at paradise.net.nz> writes
> >In message <sCLBGvAnbNL+EwtX at borve.demon.co.uk> you write:
...
> >In which case it is trivial to calculate exact powers of a surd.
> >No, I think it is not unique?
> 
> Apologies if I am missing something, or have misunderstood something,
> but have you got an example of a positive number x other than phi that
> meets the criteria? I was keeping to Leroy's definition:
>
Thanks Neil F..
Sorry, I haven't got an example.

I would like to amend my comment.

It is far easier to calculate powers...  (2+sqrt3)^n
and obtain corresponding results  than log(base 2+sqrt3)of(m).

> >>>>>Consider the integers m, m >= 2, such that:
> >>>>>ln(m) is closer to an integer than any previous term of the
> >>>>>sequence
> 
> but generalising to allow the log base to be x.
> ...
> Neil Fernandez
> 
don.mcdonald.