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

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, , is a Sum of distinct powers of 3 for . This is equivalent to the requirement that the ternary expansion of always contains a 2. This has been verified by Vardi (1991) up to . N. J. A. Sloane has conjectured that any Power of 2 has a 0 in its ternary expansion (Vardi 1991, p. 28).

**References**

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 in Base Three.'' *Computational Recreations in Mathematica.*
Reading, MA: Addison-Wesley, pp. 20-25, 1991.

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

© 1996-9

1999-05-26