Zero

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. $0^0$ is undefined, but defining $0^0=1$ allows concise statement of the beautiful analytical formula for the integral of the generalized Sinc Function

$$\int_0^\infty {\sin^a x\over x^b}\,dx = {\pi^{1-c}(-1)^{\lef... ...}\right\rfloor -c} (-1)^k{a\choose k}(a-2k)^{b-1}[\ln(a-2k)]^c$$

given by Kogan, where $a\geq b>c$, $c\equiv a-b\ \left({{\rm mod\ } {2}}\right)$, and $\left\lfloor{x}\right\rfloor$ is the Floor Function.

The following table gives the first few numbers $n$ such that teh decimal expansion of $k^n$ contains no zeros, for small $k$. The largest known $n$ for which $2^n$ contain no zeros is 86 (Madachy 1979), with no other $n\leq 4.6\times 10^7$ (M. Cook), improving the $3.0739\times 10^7$ limit obtained by Beeler et al. (1972). The values $a(n)$ such that the positions of the right-most zero in $2^{a(n)}$ 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 $2^{781,717,865}$ occurs at the 217th decimal place, the farthest over for powers up to $2.5\times 10^9$.

$k$	Sloane	$n$ such that $k^n$ 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 $n$ such that $k^n$ contains no zeros for $k=2$, 3, ... is then given by 86, 68, 43, 58, 44, 35, 27, 34, 0, 41, ... (Sloane's A020665).

See also 10, Naught, Negative, Nonnegative, Nonzero, One, Positive, Two

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.