A Figurate Number of the form

(1) |

(2) |

The only numbers which are simultaneously Square and Tetrahedral are
, , and
(giving , , and ), as proved by Meyl
(1878; cited in Dickson 1952, p. 25). Numbers which are simultaneously Triangular and
tetrahedral satisfy the Binomial Coefficient equation

(3) |

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, p. 59, 1987.

Beukers, F. ``On Oranges and Integral Points on Certain Plane Cubic Curves.''
*Nieuw Arch. Wisk.* **6**, 203-210, 1988.

Conway, J. H. and Guy, R. K. *The Book of Numbers.* New York: Springer-Verlag, pp. 44-46, 1996.

Dickson, L. E. *History of the Theory of Numbers, Vol. 2: Diophantine Analysis.* New York: Chelsea, 1952.

Guy, R. K. ``Figurate Numbers.'' §D3 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 147-150, 1994.

Meyl, A.-J.-J. ``Solution de Question 1194.'' *Nouv. Ann. Math.* **17**, 464-467, 1878.

Sloane, N. J. A. Sequence
A000292/M3382
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26