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The Base 3 method of counting in which only the digits 0, 1, and 2 are used. These digits have the following multiplication table.

$\times$ 0 1 2
0 0 0 0
1 0 1 2
2 0 2 11

Erdös and Graham (1980) conjectured that no Power of 2, $2^n$, is a Sum of distinct powers of 3 for $n>8$. This is equivalent to the requirement that the ternary expansion of $2^n$ always contains a 2. This has been verified by Vardi (1991) up to $n=2\cdot 3^{20}$. N. J. A. Sloane has conjectured that any Power of 2 has a 0 in its ternary expansion (Vardi 1991, p. 28).

See also Base (Number), Binary, Decimal, Hexadecimal, Octal, Quaternary


Erdös, P. and Graham, R. L. Old and New Problems and Results in Combinatorial Number Theory. Geneva, Switzerland: L'Enseignement Mathématique Université de Genève, Vol. 28, 1980.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 10-11, 1991.

Vardi, I. ``The Digits of $2^n$ in Base Three.'' Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 20-25, 1991.

mathematica.gif Weisstein, E. W. ``Bases.'' Mathematica notebook Bases.m.

© 1996-9 Eric W. Weisstein