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Base 2

Palindromes in bases 2 and 10.
The main source for this table comes from the following weblink.
Binary/Decimal Palindromes by Charlton Harrison (email) from Austin, Texas.
See also Sloane's sequences A007632 and A046472.

" I just finished writing a distributed client/server program for finding these numbers, and I currently
have it running on 4 different machines at the same time and I'm finding them A LOT faster. That's how
I was able to come up with those new ones. I'd say there are more to come in the near future, too."

[ December 1, 2001 ] Charlton Harrison found a new record binarydecimal palindrome 110 0010111100 0010101010 1101000011 1010000010 0000101110 0001011010 1010100001 1110100011 7475703079870789703075747 The binary string contains 83 digits !

[ April 11, 2003 ]
Dw (email) wrote me the following :

"Using a backtracking solver, I have found larger numbers.
The first of these, which is the next after the one mentioned above, is
50824513851188115831542805 (86 bit, 3*5*11*11*17*17*83*11974799*97488286319).
There are no other 86 bit double palindromes.

I also have found two 89 bit double palindromes, but I am not sure if they
are the next ones. There may be some lower ones in 89 bit (haven't searched
that space completely yet) or in 87 bit that I haven't found.
These numbers are
532079161251434152161970235 (5*29*85839676103*42748430836381)
and
552963956270141072659369255 (5*7*7*71*441607*71984077228507867)
As can be seen by their factor representation, they are all composite."

50824513851188115831542805 {26}
10101000001010100000100110101010101001100000000110010101010101100100000101010000010101 {86}

532079161251434152161970235 {27}
11011100000100000001001010010000011001110101010101110011000001001010010000000100000111011 {89}

552963956270141072659369255 {27}
11100100101100110101011000010001001111100100100100111110010001000011010101100110100100111 {89}

[ April 13, 2003 ]
Dw (email) found four new ones including the 6^th double palindromic prime :

"They are, in opposite sorted order:
138758321383797383123857831 (87 bit, composite, the only 87 bit double palindrome)
390714505091666190505417093 (89 bit, prime)
351095331428353824133590153 (89 bit, composite)
795280629691202196926082597 (90 bit, composite, not sure if this is the next one)."

When asking Dw for an explanation or description in a few words
what he meant by 'Using a backtracking solver' he replied :

"A backtracking solver is one that solves a problem made up of smaller
problems by trying every one except if it knows its earlier guesses make all
later ones depending on them impossible.

For instance, if you're in a maze and know that all corridors with green
walls eventually lead to a dead end, you can turn around as soon as you find
such a wall (backtrack) if you're trying to find the exit.

The smaller problem is avoiding a dead end, and the larger one is
finding the exit.

If you want the details:

My general strategy goes that to find a double palindromic number, the
solution to its palindromic decimal representation minus its binary
representation must be zero (since they are equal).

Furthermore, you can write a decimal palindrome like 101 * a + 10 * b, where
a and b are digits. The same can be done for binary, and you end up with a
giant (linear diophantine) equation of positive decimal and negative binary factors.

The factors can then be solved, one at a time, using the extended euclidean
algorithm. These are the smaller problems.

I also have a table of maximum and minimum values for each step. That way, if
it's impossible for the binary factors left to subtract enough from the
decimal factors to get zero (or the other way around), the solver backtracks."

[ April 13, 2003 ] Dw found a new record binarydecimal palindrome 1010010001 1101011100 1110010101 0010100001 0100000010 1000010100 1010100111 0011101011 1000100101 795280629691202196926082597 The binary string contains 90 digits ! The decimal string contains 27 digits !

138758321383797383123857831 {27}
111001011000111001101111010110010111011100111001110111010011010111101100111000110100111 {87}

351095331428353824133590153 {27}
10010001001101011010101000000110111100000100000100000111101100000010101011010110010001001 {89}

390714505091666190505417093 {27}
10100001100110001000000111100001100011000111011100011000110000111100000010001100110000101 {89}

795280629691202196926082597 {27}
101001000111010111001110010101001010000101000000101000010100101010011100111010111000100101 {90}

[ May 21, 2003 ]
Dw (email) found a new Binary/Decimal Palindrome :

"I have been trying to find a polynomial time (growth of time needed is a
polynomial of number of bits) algorithm for finding double palindromes of
base 2 and 10. I haven't succeeded yet (my backtracking solver being
exponential), but I have found some interesting things.

For one, to find double palindromes in base 2 and 8 is very simple. Each base 8
digit maps to three bits. Therefore, every double palindrome must consist
of digits who themselves are double palindromes.

For instance, 757 as well as 575 is double palindromic. (These values are 495
and 381 in decimal respectively).
5 maps to 101 in binary, and 7 to 111 in binary.

I have found 1609061098335005338901609061 (91 bits composite),
and this is the only 91 bit one. No 92 bit double palindrome exists, and
93 bits seems to require several days of searching; therefore I'm trying to
find a polynomial time algorithm as mentioned above.

Another approach could be to create a networked version (to do the search on
multiple computers), but I haven't done that yet."

[ May 21, 2003 ] Dw found a new record binarydecimal palindrome 1 0100110010 1111101111 1100001110 1100100000 0010001000 0000100110 1110000111 1110111110 1001100101 1609061098335005338901609061 The binary palindrome contains 91 digits ! The decimal string contains 28 digits !

1609061098335005338901609061 {28}
1010011001011111011111100001110110010000000100010000000100110111000011111101111101001100101 {91}

[ June 12, 2003 ]
Dw (email) found new Binary/Decimal Palindromes :

"I rewrote my program to use another strategy at finding the numbers, and this
let me search somewhat faster. As a result, I have found binary/decimal
palindromes up to 102 bits -- broken the 100 bit barrier so to speak.

These are:

None at 92 or 93 bits.
17869806142184248124160896871 (94 bits)
19756291244127372144219265791 (94 bits)
30000258151173237115185200003 (95 bits)
30658464822225352222846485603 (95 bits)
56532345659072227095654323565 (96 bits)
None at 97 or 98 bits.
378059787464677776464787950873 (99 bits)
1115792035060833380605302975111 (100 bits)
None at 101 bits.
3390741646331381831336461470933 (102 bits)

There may be higher ones at 102 bits; I haven't completed the search there."

[ June 17, 2003 ] Dw found a new record binarydecimal palindrome 10 1010110011 0000010011 1100000010 0001000111 1010010111 1010010111 1000100001 0000001111 0010000011 0011010101 3390741646331381831336461470933 The binary palindrome contains 102 digits ! The decimal string contains 31 digits !

Binary/Decimal Palindromes by Charlton Harrison (email) : the longest list in existence ?

Now that we entered the world of binary numbers let's find out more about Palindromic Squares in base 2.
I retrieved these from Sloane's table namely entry number A003166 !

Base 3 and base 9

You all probably noticed that I 'restrict' myself searching for large palindromes mainly in the decimal numbersystem.
Not so for Kevin Brown ! Here are a few palindromic tetrahedrals that he discovered in other base representations.

Base 12

Alain Bex (email) gave also some palindromic squares in base 12 :

Note that again some of the basenumbers are palindromic themselves
and share many similarities with the palindromic squares in base 10 !
See sequence with index number A029737 in Sloane's table for these numbers.

Base 16

[ December 28, 2008 ]
Matt S.
asked himself how difficult is it to generate the elements of these sequences ?

Numbers n such that n^3 is palindromic in base 16.

Palindromic cubes in base 16.

as he has derived e.g. 1152921504606846977 that's well beyond what
you have listed.

Any additional links/info you can give me would be appreciated.

Here is my reply after some perusing (PDG):

I found

You added

Is this how you derived the number by working backwards from
the pattern 1_0_[x]_3_0_[x]_3_0_[x]_1 you discovered ?
Anyway well done and congratulations!

Ten years ago I submitted my sequences. So forgive me that I didn't
kept my original code. But from recollection I just searched in
a straightforward manner with the UBASIC program from zero till
I got tired with the last entry. Modern fast computers on the other
hand should have no problem in recreating the known sequences numbers
from A029735 & A029736.
Of course your number cannot yet be added as there might be other
solutions in between very probably.

Matt S. wrote :

Thanks for the reply.

The largest number I have right now is approx 281 bits. I have *many*
more too. Apologies for the rudimentary notation - this is all fairly
new to me. I'm actually generating these numbers with a friend, all
coming from a coded letter that was sent to fermilab
http://www.symmetrymagazine.org/breaking/2008/05/15/code-crackers-wanted/
I have little to no training in this area.

1_0_[x]_3_0_[x]_3_0_[x]_1 is correct for the power of 3

It can be extended into other powers as well. For example :

I'd like to track down where the aforementioned letter came from - and
how this sequence is involved. Surely not a coincidence ? Any ideas ?
I'm eager to hear what your thoughts are regarding the letter / sequence.
As I said, I realize it's an odd request... so any insight is welcome.
If you need more data, please feel free to ask.

Binary/Decimal Palindromes by Charlton Harrison (email) : the longest list in existence ?

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences I sampled the following base X palindromic numbers sequences from the table : %N Binary expansion is palindromic. under A006995 -- Sum of digits A043261 %N Palindromes in base 3 (written in base 10). under A014190 -- Sum of digits A043262 %N Palindromes in base 4 (written in base 10). under A014192 -- Sum of digits A043263 %N Palindromic in base 5. under A029952 -- Sum of digits A043264 %N Palindromic in base 6. under A029953 -- Sum of digits A043265 %N Palindromic in base 7. under A029954 -- Sum of digits A043266 %N Palindromic in base 8. under A029803 -- Sum of digits A043267 %N Palindromic in base 9. under A029955 -- Sum of digits A043268 %N Palindromes. under A002113 -- Sum of digits A043269 %N Palindromic in base 11. under A029956 -- Sum of digits A043270 %N Palindromic in base 12. under A029957 -- Sum of digits A043271 %N Palindromic in base 13. under A029958 -- Sum of digits A043272 %N Palindromic in base 14. under A029959 -- Sum of digits A043273 %N Palindromic in base 15. under A029960 -- Sum of digits A043274 %N Palindromic in base 16. under A029730 -- Sum of digits A043275 %N Palindromic in bases 2 and 3. under A060792. %N Palindromic in bases 2 and 10. under A007632. %N Palindromic in bases 3 and 10. under A007633. %N Palindromic in bases 4 and 10. under A029961. %N Palindromic in bases 5 and 10. under A029962. %N Palindromic in bases 6 and 10. under A029963. %N Palindromic in bases 7 and 10. under A029964. %N Palindromic in base 8 and base 10. under A029804. %N Palindromic in bases 9 and 10. under A029965. %N Palindromic in bases 11 and 10. under A029966. %N Palindromic in bases 12 and 10. under A029967. %N Palindromic in bases 13 and 10. under A029968. %N Palindromic in bases 14 and 10. under A029969. %N Palindromic in bases 15 and 10. under A029970. %N Square in base 2 is a palindrome. under A003166. %N Squares which are palindromes in base 2. under A029983. %N n^2 is palindromic in base 3. under A029984. %N Squares which are palindromic in base 3. under A029985. %N n^2 is palindromic in base 4. under A029986. %N Squares which are palindromic in base 4. under A029987. %N n^2 is palindromic in base 5. under A029988. %N Squares which are palindromic in base 5. under A029989. %N n^2 is palindromic in base 6. under A029990. %N Squares which are palindromic in base 6. under A029991. %N n^2 is palindromic in base 7. under A029992. %N Squares which are palindromic in base 7. under A029993. %N n^2 is palindromic in base 8. under A029805. %N n in base 8 is a palindromic square. under A029806. %N n^2 is palindromic in base 9. under A029994. %N Squares which are palindromic in base 9. under A029995. %N Square is a palindrome. under A002778. %N Palindromic Squares. under A002779. %N n^2 is palindromic in base 11. under A029996. %N Squares which are palindromic in base 11. under A029997. %N n^2 is palindromic in base 12. under A029737. %N Squares which are palindromic in base 12. under A029738. %N n^2 is palindromic in base 13. under A029998. %N Squares which are palindromic in base 13. under A029999. %N n^2 is palindromic in base 14. under A030072. %N Squares which are palindromes in base 14. under A030074. %N n^2 is palindromic in base 15. under A030073. %N Squares which are palindromes in base 15. under A030075. %N n^2 is palindromic in base 16. under A029733. %N Palindromic squares in base 16. under A029734. %N n^3 is palindromic in base 4. under A046231. %N Cubes which are palindromes in base 4. under A046232. %N n^3 is palindromic in base 5. under A046233. %N Cubes which are palindromes in base 5. under A046234. %N n^3 is palindromic in base 6. under A046235. %N Cubes which are palindromes in base 6. under A046236. %N n^3 is palindromic in base 7. under A046237. %N Cubes which are palindromes in base 7. under A046238. %N n^3 is palindromic in base 8. under A046239. %N Cubes which are palindromes in base 8. under A046240. %N n^3 is palindromic in base 9. under A046241. %N Cubes which are palindromes in base 9. under A046242. %N Cube is a palindrome. under A002780. %N Palindromic cubes. under A002781. %N n^3 is palindromic in base 11. under A046243. %N Cubes which are palindromes in base 11. under A046244. %N n^3 is palindromic in base 12. under A046245. %N Cubes which are palindromes in base 12. under A046246. %N n^3 is palindromic in base 13. under A046247. %N Cubes which are palindromes in base 13. under A046248. %N n^3 is palindromic in base 14. under A046249. %N Cubes which are palindromes in base 14. under A046250. %N n^3 is palindromic in base 15. under A046251. %N Cubes which are palindromes in base 15. under A046252. %N n^3 is palindromic in base 16. under A029735. %N Cubes which are palindromes in base 16. under A029736. %N Palindromic primes in base 2. under A016041. %N Palindromic primes in base 3. under A029971. %N Palindromic primes in base 4. under A029972. %N Palindromic primes in base 5. under A029973. %N Palindromic primes in base 6. under A029974. %N Palindromic primes in base 7. under A029975. %N Palindromic primes in base 8. under A029976. %N Octal palindromes which are also primes. under A006341. %N Palindromic primes in base 9. under A029977. %N Palindromic primes. under A002385. %N Palindromic primes in base 11. under A029978. %N Palindromic primes in base 12. under A029979. %N Palindromic primes in base 13. under A029980. %N Palindromic primes in base 14. under A029981. %N Palindromic primes in base 15. under A029982. %N Palindromic primes in base 16. under A029732. %N Palindromic primes in base 10 and base 2. under A046472. %N Palindromic primes in base 10 and base 3. under A046473. %N Palindromic primes in base 10 and base 4. under A046474. %N Palindromic primes in base 10 and base 6. under A046475. %N Palindromic primes in base 10 and base 7. under A046476. %N Palindromic primes in base 10 and base 8. under A046477. %N Palindromic primes in base 10 and base 9. under A046478. %N Palindromic primes. under A002385. %N Palindromic primes in base 10 and base 11. under A046479. %N Palindromic primes in base 10 and base 12. under A046480. %N Palindromic primes in base 10 and base 13. under A046481. %N Palindromic primes in base 10 and base 14. under A046482. %N Palindromic primes in base 10 and base 15. under A046483. %N Palindromic primes in base 10 and base 16. under A046484. %N Palindromic primes in bases 2 and 4. under A056130. %N Palindromic primes in bases 2 and 8. under A056145. %N Palindromic primes in bases 4 and 8. under A056146. %N Not palindromic in any base from 2 to n-2. under A016038. %N Smallest palindrome greater than n in bases n and n+1. under A048268. %N First palindrome greater than n+2 in bases n+2 and n. under A048269. %N The first non-trivial (k>n+2) palindromic prime in both bases n and n+2. under A057199. %N Symmetric bit strings (bit-reverse palindromes), including as many leading as trailing zeros. under A057890. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

1. A016038 Strictly non-palindromic numbers: n is not palindromic in any base b with 2 <= b <= n-2. - N. J. A. Sloane.
2. A100563 Number of bases less than sqrt(n) in which n is a palindrome. - Gordon Robert Hamilton

Kevin Brown informed me that he has more info about tetrahedral palindromes in other base representations.
Link to his article :
On General Palindromic Numbers

Alain Bex (email) sent me the first palindromic squares in base 12 - go to topic.

Dw (email) found several binary/decimal palindromes of record lengths - go to topic.

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(All rights reserved) - Last modified : May 1, 2013.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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