*Post by a***@verizon.net*2004595370006815987563563

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.net*The 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.net*The 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.net*The 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.