The Integer denoted 0 which, when used as a counting number, means that no objects are present. It is the only Integer (and, in fact, the only Real Number) which is neither Negative nor Positive. A number which is not zero is said to be Nonzero.

Because the number of Permutations of 0 elements is 1, 0! (zero Factorial) is often defined as
1. This definition is useful in expressing many mathematical identities in simple form. A number *other than 0* taken
to the Power 0 is defined to be 1. is undefined, but defining allows concise statement of the beautiful
analytical formula for the integral of the generalized Sinc Function

given by Kogan, where , , and is the Floor Function.

The following table gives the first few numbers such that teh decimal expansion of contains no zeros, for small .
The largest known for which contain no zeros is 86 (Madachy 1979), with no other
(M. Cook),
improving the
limit obtained by Beeler *et al. *(1972). The values such that the positions of the
right-most zero in increases are 10, 20, 30, 40, 46, 68, 93, 95, 129, 176, 229, 700, 1757, 1958, 7931, 57356,
269518, ... (Sloane's A031140). The positions in which the right-most zeros occur are 2, 5, 8, 11, 12, 13, 14, 23, 36, 38,
54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, ... (Sloane's A031141). The right-most zero of
occurs at
the 217th decimal place, the farthest over for powers up to
.

Sloane | such that contains no 0s | |

2 | Sloane's A007377 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, ... |

3 | Sloane's A030700 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, ... |

4 | Sloane's A030701 | 1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43, ... |

5 | Sloane's A008839 | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58, ... |

6 | Sloane's A030702 | 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44, ... |

7 | Sloane's A030703 | 1, 2, 3, 6, 7, 10, 11, 19, 35 |

8 | Sloane's A030704 | 1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27 |

9 | Sloane's A030705 | 1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34 |

11 | Sloane's A030706 | 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 41, ... |

While it has not been proven that the numbers listed above are the only ones without zeros for a given base, the probability that any additional ones exist is vanishingly small. Under this assumption, the sequence of largest such that contains no zeros for , 3, ... is then given by 86, 68, 43, 58, 44, 35, 27, 34, 0, 41, ... (Sloane's A020665).

**References**

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT
Artificial Intelligence Laboratory, Memo AIM-239, Item 57, Feb. 1972.

Kogan, S. ``A Note on Definite Integrals Involving Trigonometric Functions.'' http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, pp. 127-128, 1979.

Pappas, T. ``Zero-Where & When.'' *The Joy of Mathematics.* San Carlos, CA: Wide World Publ./Tetra, p. 162, 1989.

Sloane, N. J. A. Sequence
A007377/M0485
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