Newsgroups: rec.puzzles From: hoey@AIC.NRL.Navy.Mil (Dan Hoey) Date: 25 Sep 92 15:37:25 GMT Subject: Re: Factorization puzzle (many more solutions) In response to b...@CS.ColoState.EDU (dave boll)'s puzzle: > Is there an integer whose reverse (written in base 10) contains > exactly the same prime factors as the original number? j...@lentil.princeton.edu (Jaroslaw Tomasz Wroblewski) answers: > One can recycle 2178 here (its reversal is 4 times itself). It is no coincidence that 2178 is a facdrome, a number that is a divisor of its reverse. Every base-ten facdrome has a recycled factorization. I found all the facdromes last year, and you can read about them in the FAQ by mailing "send palintiples" to net...@peregrine.com. (A palintiple is the reverse of a facdrome.) In short, we can characterize the facdromes as follows. For any language L, let PalNLZ(L) denote the set of palindromes over the (infinite) alphabet (L union 0*), except that we exclude from PalNLZ(L) those strings beginning or ending with zero. Then the set of facdromes is the union of PalNLZ(219*78) and PalNLZ(109*89). For example, the following seven numbers are facdromes: 21978, 1089, 1099989, 219978 219978, 10989 0000 10989, 2178 0 21978 21978 0 2178, 10989 1089 000 1089 10989. In addition, I have found some non-facdromes with recycled factorizations. The set PalNLZ(439*56), containing numbers that are two-thirds of their reverse, has this property. Examples include 439956 and 435604356. More interesting is the set PalNLZ(329*67), containing numbers that are three-seventh of their reverse. These numbers have recycled factorizations if and only if they are multiples of seven. Examples include 32999967, 326732673267, 3267000003267, 32967000032967, and 329967000329967. I have not found any other numbers with recycled factorizations, but I would not be surprised if they exist. Also, in bases other than ten, I suspect there may be facdromes that do not have recycled factorizations, but I have not gone looking for them. Dan Hoey Hoey@AIC.NRL.Navy.Mil