Martin Gardner by Scientific American Editors

Author:Scientific American Editors
Language: eng
Format: epub
Publisher: The Rosen Publishing Group, Inc
Published: 2014-09-12T00:00:00+00:00

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Hexagonal numbers as the difference between consecutive cubes

Illustration by Ilil Arbel

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Apart from a hex of 1, the first triangular hex is 91 and the first square hex is 169. Readers who know the Pellian technique may enjoy searching for recursive procedures that will generate each of these infinite sequences of numbers and for their nonrecursive formulas. The Pellian for square hexes is 3x2 + 1 y2, which is solved by finding the convergents of the continued fraction for the square root of 3. The next square hex after 169 is 32761, and the next is 6355441. Are there hexes that are both square and triangular? Is there a cubical hex?

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