olson@ucbcory.BERKELEY.EDU (Nels Olson) (10/06/85)
Here, for your enjoyment, is a curiosity I stumbled upon
several years ago which has interested me ever since.
(For the record, this is my first posting, so be tolerant.)
I was learning how to program and was running short of
interesting programs to write, so I asked my mother for ideas,
and she came up with the following:
First she thought of this particular case: 5 squared is 25,
the digits of 25 multiplied together (2*5) is 10, and that
result (10) is twice 5. So I got to write a program
(what fun!) that would find all the integers X such that the
product of the digits of X*X is equal to 2X.
As I wrote the program I slightly modified the problem:
if a digit in the square was 0, it got ignored--i.e. the program
calculated the product of the non-zero digits of X*X, and
compared it with 2X.
The program found these numbers: 2, 5, 54, 648, and 2160.
The curiosity: All the "non-trivial" (read "having more than
one digit", perhaps) solutions are divisible by 54. Why? I
don't know.
Here are some more fun facts: Since my mother's name is
Faith, and in honor of the facts that:
(1) she invented the problem,
(2) she was 54 years old when she did so,
(3) she was 648 months old when she did so, and
(4) it was the first name I thought of,
I call these five (and any other such) numbers 'Faithy numbers.'
Since then I have run some more programs that searched for
Faithy numbers; there are no others less than 12 million.
I have also played with two similar problems. The first is
with the 'dig-prod' being other multiples of X besides 2.
I call any number which divides the dig-prod of its square a
"general Faithy number." (I have a list of the first few
general Faithy numbers. There happen to be exactly 54 of them
(there's that number again) less than 10 million, another
meaningless coincidence. The largest one I know of is
60963840, which is 'predictable' (I'll let you figure out
how) from the 51st one, 6096384.) The second related problem
is with other bases besides 10.
A few results for general Faithy numbers: there are none
less than 2 to the 200th power in base 3. There are none
(besides 1, which works in any base) less than 50000 in base 4.
The only ones less than 50000 in base 8 are 1, 2 and 21
(i.e. 21=25 base 8). There are quite a few of them in bases
5,6,7,9 and 10.
Does anyone have any thoughts on the likelihood that the
divisibility-by-54 curiosity is coincidental? Or on the
relation of this problem to the meaning of life? Better,
could anyone either prove that divisibility by 54 is
necessary for Faithy-number-hood, or find a counterexample?
My guess is that it is not necessary but that counterexamples
are rare.
---
Nels Olson
(Sorry, but I don't know how to reach me by e-mail.
Maybe you can figure it out from the header.)
Preacher: "You're all different."
Member of audience: "I'm not!" -- M.P.