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 1921-1926 (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.

# Web-books 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

Articles of Doron Zeilberger