Post by a***@verizon.net2004595370006815987563563
Challenged, not dared. I thought I'd find it with clever programming
and a slow computer before you'd find it with simpler programming and
a fast computer.
I didn't realize that you had read my posts explaining how I'd speed
up the search. Nor did I realize that life would intervene, and I
wouldn't have a chance to even start writing the program for several
days.
However, I have some new ideas on how to speed up the search even
more. You found the 8th dual binary-ternary palindrome, but I think
I'll find the 9th, even though my computers are lot older and slower
than yours.
Post by a***@verizon.netThe above number was, reluctantly, confirmed by Keith so I'm fairly
sure it's a solution.
"Reluctantly"? I don't spend all my time online, unlike some people.
Also, it took me time to confirm it. After I got your email, I
quickly wrote a base conversion program that will work on numbers of
*any* length, rather than being limited to my machine's word size.
2: 110101000011111010101010100101111011110111011110111101001010101010111110000101011
3: 221010112100202002120002212200021200202001211010122
And while I'm at it,
4: 12220133111110233132323313221111113300223
5: 32102202104224200013133120304013223
6: 13022330515021132403133213433455
7: 42335264631355134314453431502
8: 650372524573673675125276053
9: 27115322076085607622054118
10: 2004595370006815987563563
11: 226978108866737944128AA5
12: 376A681B84083816479B88B
13: 81628CAA5080999952070B
14: 19D5D13A7B59A584D9B839
15: 6065C5B980B41BE6695C8
16: 1A87D552F7BBDE9557C2B
17: 8693G29C68E42CB3923G
18: 2EH0GH751B467B00629H
19: 104EE4616ED4017HG43E
20: 7CIF7I6A24B4JG258I3
21: 33F58D8E80EJD070HE2
22: 185EJJ52KFFIF089855
23: E4K2GJ372F7I20793B
24: 6LAEJLNLK81FAHKN4B
25: 3B2AB4CM01GI7341HD
26: 1JP4502L17D1P91I0B
27: P3E9K2F2NI7IK1M3H
28: E16J3QRQ9O6HPRJFN
29: 808NK3HD8OCIB9296
30: 4JL39PB49D6GI6NSN
31: 2ND8FQGIN1227F3JA
32: 1L1ULABRRNNKLAV1B
33: 10ELFD1DSNJ0JJFT5
34: LCLUSEOWPM8ASQ1X
35: DT8WCK2S00ECFAJN
36: 92FIVUD9G39K9RMZ
Post by a***@verizon.netThe code is pretty messy as I had been trying a number of different
things. it requires the GMP library and assumes a modern 64 bit
programming environment.
Mine, when I get it done, will work even on an eight-bit machine, as I
store one digit per array element.
Post by a***@verizon.netThe code is embarrassingly parallel so a much larger machine could
search more space faster, ...
My approach is to shrink the search space.
Is your number, 2004595370006815987563563, prime? It's not divisible
by any small divisor. None of the other known solutions are prime.
There appears to be no pattern to the divisibility, except that none
are divisible by 2 or 3.
Last night I explored dual palindromes in other pairs of bases. Here
are how many non-trivial dual palindromes there below 100 trillion
(10^14) in each pair of bases 2 though 16. (I define a trivial dual
palindrome as one that's a single digit in one or both bases. For
instance 1 though 7 are all trivial dual octal-decimal palindromes.)
2 3 4 5 6 7 8 9 10 11 12 13 14 15
3 4
4 6142 12
5 13 16 18
6 40 36 31 19
7 23 33 25 36 19
8 33276 16 3080 28 32 25
9 30 9716 32 18 25 40 29
10 42 38 35 30 66 37 39 39
11 25 32 29 51 19 37 31 36 51
12 29 34 41 35 36 35 41 31 26 76
13 26 34 31 31 41 22 44 46 47 37 94
14 58 32 43 35 27 30 46 45 31 46 35 109
15 25 54 28 44 43 60 25 34 41 43 40 44 133
16 6778 22 9210 38 43 27 308 36 46 42 41 40 44 138
2 3 4 5 6 7 8 9 10 11 12 13 14 15
Note that:
* There are dual palindromes for every pair of bases
* Binary-ternary dual palindromes are by far the rarest
* Two bases having a factor in common seems to have little or no
effect unless they're powers of the same number
The 72252 non-trivial dual palindromes include 541 triple palindromes,
38 quadruples, and one quintuple: 11111111 (2), 3333 (4), 313 (9), 212
(11), FF (16).
A triple that contains a base 10 number is 1221010220101221 (3),
24043234042 (5), 27711772 (10).
I suspect that for every pair of bases, there are infinitely many dual
palindromes. And that for nearly every triple of bases, there are
finitely many triple palindromes, usually a very small number.
There are special cases where there are infinitely many triple
palindromes. For instance 64^N-1 is a palindrome in bases 2, 4,
and 8 for all N.
--
Keith F. Lynch - http://keithlynch.net/
Please see http://keithlynch.net/email.html before emailing me.