🚀 go-pugleaf

Stephen Whitfield (stphw@ozemail.com.au) wrote:
: mmeiss@beavis.indiana.edu (Mark Meiss) wrote:

: >In article <4doa5n$4uv@tkhut.sojourn.com>,
: >Rod Shelton <roshne@faerun.sojourn.com> wrote:
: >>I have an honors math section this term, and we are encouraged to seek 
: >>discussion on the problems we are given.  
: >>
: >>Problem 1:
: >>Does there exist a positive integer N such that 2 < N < 2x10^14
: >>and such that N is a perfect square, cube and fifth power?
: >>Consider any generalizations which may arise.

: >Sure, N exists.  You want N^(1/2), N^(1/3), N^(1/5) to be integers.
: >So pick N = x^(2*3*5) = x^30.  Then N^(1/2) = x^15, N^(1/3) = x^10,
: >N^(1/5) = x^6.  The generalization follows from there... the set of
: >numbers for which this holds is all x such that 2 < x^30 < 2x10^14.
: >--Mark
:  
: But x^15, x^10 and x^6 do not satisfy the question.?.?
: Hmmm   : Stephen W.

Sure they do, for the appropriate value(s) of x.
x=0,1 both fail since the resulting N is not greater than 2
How about x=2,3,?  

Obvious (easy) Generalization:
Can N be found which is a square, cube, fifth AND seventh power?
If so, how large is the first one?
How many such N are there?

More difficult:
What percent of the positive integers are neither square, nor cube, nor 
any higher power?
(More carefully stated:  The Limit as M goes to infinity of the percent 
of positive integers from 1 to M which cannot be written as a power 
larger than 1.)

 _
|/   |\/|         ||  Fermat is dead!  ||  rob_morewood@panam.wimsey.bc.ca
|\ob |  |orewood  || Long Live Wiles!! ||    Beautiful British Columbia