zeroless squares

[Joshua Zucker]

On Mon, 28 Feb 2005 08:53:33 +0100, Pfoertner, Hugo
<Hugo.Pfoertner at muc.mtu.de> wrote:
> Smallest power with base>1, exponent>1 whose decimal representation doesn't
> contain the digit 0:
> 
> base exponent                base^exponent
> 12 20               3833759992447475122176
I add a term here:
381 21 1582794342217312156827221746448942623537121214738891981

> 22 22       341427877364219557396646723584
> x^40 exceeds 33 decimal digits.

For x^40, if there is a term, the base exceeds 1000000, anyway, and
I'll improve that bound shortly.

Beyond x^40, I have:
11^41 = 4978518112499354698647829163838661251242411
max 10000 reached, none found
4^43 = 77371252455336267181195264
6^44 = 17324272922341479351919144385642496
max 10000 reached, none found
max 10000 reached, none found
max 10000 reached, none found
max 10000 reached, none found
2^49 = 562949953421312
max 10000 reached, none found

which is definitely suggesting that if it's not a small base, there's
unlikely to be one for QUITE a long time.  For the exponents 50-100,
my quick checks find only bases of 2 or 3, that work, and no other
bases < 10000.

--Joshua Zucker