**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.