Zeroless Arithmetic: Representing Integers ONLY using ONE
By
Edinah K. Gnang and Doron Zeilberger
.pdf
.ps
.tex
Written: March 3, 2013
[Appeared in J. Difference Equations and Applications
Volume 19, Issue 11, November 2013, pages 19211926 (DOI: 10.1080/10236198.2013.791288)]
Suppose that you have a Reverse Polish Calculator where the only keys left
are the "plus", "times", and "power", "1", and, of course the "Enter" key.
In how many ways can you express 40?
(Ans.: 2601671905509333123020 ways).
Also, how to generate, uniformly at random,
one such expression?
Also, what is the shortest way of doing it?
Also, what if you can only use addition and multiplication? etc. etc.
If you always wanted to know the answers to these fascinating questions, this article,
(and especially the Maple package ArithFormulas)
are for you! But even if you couldn't care less,
the methodology of experimental math presented here may benefit you
for problems that you do care about.
Maple Package
Webbooks from the Maple package ArithFormulas

If you want to see the number of arithmetical formulas only using
addition and multiplication for n from 1 to 40, as well as
shortest formulas for n between 1 and 8000,
the
input file
would yield the
output file

If you want to see the number of arithmetical formulas only using
addition, multiplication and exponentiation for n from 1 to 40, as well as
shortest formulas for n between 1 and 8000,
the
input file
would yield the
output file

If you want to see the number of arithmetical formulas only using
addition and exponentiation for n from 1 to 40, as well as
shortest formulas for n between 1 and 8000,
the
input file
would yield the
output file
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