Zerofree

An integer whose decimal digits contain no zeros is said to be zerofree. The first few positive zerofree integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ... (OEIS A052382).

Zerofree squares are easy to generate, e.g.,

(1)

Around 1990, D. Hickerson considered the problem of finding large zerofree cubes. After some experimentation, he found a formula that generated infinitely many of them. In March 1998, Bill Gosper asked about 0-free th powers, pointing out that heuristically we should expect there to be infinitely many zerofree squares, cubes, ..., 21st powers, but only finitely many 22nd powers, etc. At this point, Hickerson couldn't locate his formula for cubes, and so came up with the new formula

(2)

which is 0-free if  and .

In April 1999, Ed Pegg conjectured on sci.math that there are only finitely many zerofree cubes, so Hickerson posted his new counterexample, (mistakenly claiming that it was the one he had found 10 years ago). A few days later, Lew Baxter posted the slightly simpler example

(3)

known as the Baxter-Hickerson function.

There is apparently no proof that there exist infinitely many zerofree 4th powers, 5th powers, ..., or 21st powers.

REFERENCES:

Sloane, N. J. A. Sequence A052382 in "The On-Line Encyclopedia of Integer Sequences."