On Numbers in Different Bases: Symmetries and a Conjecture

ABSTRACT

Any number may be written in many different ways, using different strings in different bases. In few, very special cases, a symmetry emerges which is usually hidden beneath the surface: 230₁₆₄ and 164₂₃₀ are both equal to 54284 in base ten. This article analyzes the solution set to the (constrained) Diophantine equation that implements such symmetry, culminating in a conjecture on the number of solutions of the equation.

KEYWORDS:

• radix representation
• diophantine equations in many variables
• counting solutions of diophantine equations

2000 AMS Subject Classification:

• 11A63
• 11D72
• 11D45

Acknowledgments

The authors wish to thank P. Natalini, who unknowingly suggested the argument during one of his lectures, F. Pappalardi for fruitful discussion on Section 3.1, and the anonymous referee for his constructive comments.

Funding

The research of C. F. was partly supported by GNFM-INdAM.

Notes

1 “Imagine a multitude of objects of the same kind assembled together; for example, a company of horsemen. One of the first things that must strike a spectator, although unused to counting, is, that to each man there is a horse. Now, though men and horses are things perfectly unlike, yet, because there is one of the first kind to everyone of the second, one man to every horse, a new notion will be formed in the mind of the observer, which we express in words, by saying that there is the same number of men as of horses.”, as taken from [De Morgan 40, p. 1].

2 A total order relation on the set of all lists can be naturally introduced by M ⩾ N⇔M_b ⩾ N_b.

3 For ease of tabulation, we chose to list our experimental results so as to have a single counting index k for the whole table.