Alternation Numbers

davar55 · 2008-10-07, 19:33

Define an "alternation number" as an integer whose (decimal)
representation contains exactly two different digit values,
appearing in alternating place values (e.g. 1212 and 37373).

Find an alternation number which factors into two (or more)
alternation numbers each containing at least three digits
(if possible).

grandpascorpion · 2008-10-10, 22:49

There's no solutions of the form: aba * cdc.

Uncwilly · 2008-10-11, 00:01

what about the forms: aba * aca , aba * cac , aba * bcb , aba * cbc ?

grandpascorpion · 2008-10-11, 02:35

None of them work.

Brian-E · 2008-10-11, 19:10

Tested all alternation numbers as far as 98,989,898,989,898,989
= 9*10^16 + 8*10^15 + ... + 9*10^0
None of the factorizations work.

Brian-E · 2008-10-11, 19:57

Oops. I have misread the puzzle.
My trivial effort above looked only at the prime factorizations of alternation numbers, but the problem does not state that the alternation factors have to be prime.
Sorry. Keep trying.

2008-10-07, 19:33	#1
davar55 May 2004 New York City 5·7·11² Posts	Alternation Numbers Define an "alternation number" as an integer whose (decimal) representation contains exactly two different digit values, appearing in alternating place values (e.g. 1212 and 37373). Find an alternation number which factors into two (or more) alternation numbers each containing at least three digits (if possible).

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Carmichael numbers and Devaraj numbers	devarajkandadai	Number Theory Discussion Group	0	2017-07-09 05:07
6 digit numbers and the mersenne numbers	henryzz	Math	2	2008-04-29 02:05
LLT numbers, linkd with Mersenne and Fermat numbers	T.Rex	Math	4	2005-05-07 08:25

2008-10-10, 22:49	#2
grandpascorpion Jan 2005 Transdniestr 503 Posts	There's no solutions of the form: aba * cdc.

2008-10-11, 00:01	#3
Uncwilly 6809 > 6502 """"""""""""""""""" Aug 2003 101×103 Posts 2·7·19·37 Posts	what about the forms: aba * aca , aba * cac , aba * bcb , aba * cbc ?

2008-10-11, 02:35	#4
grandpascorpion Jan 2005 Transdniestr 503₁₀ Posts	None of them work.

2008-10-11, 19:10	#5
Brian-E "Brian" Jul 2007 The Netherlands 6305₈ Posts	Tested all alternation numbers as far as 98,989,898,989,898,989 = 910^16 + 810^15 + ... + 9*10^0 None of the factorizations work.

2008-10-11, 19:57	#6
Brian-E "Brian" Jul 2007 The Netherlands CC5₁₆ Posts	Oops. I have misread the puzzle. My trivial effort above looked only at the prime factorizations of alternation numbers, but the problem does not state that the alternation factors have to be prime. Sorry. Keep trying.