Palindromic Numbers other than Base 10

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Introduction

Palindromic numbers are numbers which read the same from

left to right (forwards) as from the right to left (backwards)

Here are a few random examples : 7, 3113, 44611644

Go directly to the Palindromic cubes in bases 2 to 17 tables
Go directly to the Base 16 topic

Palindromic cubes in bases 2 to 17

Index Nr
Decimal equivalent of numbers whose
cube in base 2 is a palindrome
L root 10
L cube 10
L base 2

Next > 10^8

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 3 is a palindrome
L root 10
L cube 10
L base 3

Next > 10^10

3 Prime!    2 1 1 2

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 4 is a palindrome
A046231 | A046232
L root 10
L cube 10
L base 4

Next > 10^10

18 4294967297 10 29 49

17 1073741825 10 28 46

16 268435457 9 26 43

15 67108865 8 24 40

14 16777217 8 22 37

13 4194305 7 20 34

12 1048577 7 19 31

11 262145 6 17 28

10 Prime!    65537 5 15 25

9 16385 5 13 22

8 4097 4 11 19

7 1025 4 10 16

6 Prime!    257 3 8 13

5 65 2 6 10

4 Prime!    17 2 4 7

3 Prime!    5 1 3 4

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 5 is a palindrome
A046233 | A046234
L root 10
L cube 10
L base 5

(Recalc) Next > 10^10

23 476837158203126 15 45 64

22 95367431640626 14 42 61

21 19073486328126 14 40 58

20 3814697265626 13 38 55

19 762939453126 12 36 52

18 152587890626 12 34 49

17 30517578126 11 32 46

16 6103515626 10 30 43

15 1220703126 10 28 40

14 244140626 9 26 37

13 48828126 8 24 34

12 9765626 7 21 31

11 1953126 7 19 28

10 390626 6 17 25

9 78126 5 15 22

8 15626 5 13 19

7 3126 4 11 16

6 626 3 9 13

5 126 3 7 10

4 26 2 5 7

3 6 1 3 4

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 6 is a palindrome
A046235 | A046236
L root 10
L cube 10
L base 6

Next > 10^10

15 2176782337 10 29 37

14 860627456 9 27 35

13 362797057 9 26 34

12 60466177 8 24 31

11 10077697 8 22 28

10 1679617 7 19 25

9 279937 6 17 22

8 46657 5 15 19

7 7777 4 12 16

6 Prime!    1297 4 10 13

5 217 3 8 10

4 Prime!    37 2 5 7

3 Prime!    7 1 3 4

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 7 is a palindrome
A046237 | A046238
L root 10
L cube 10
L base 7

Next > 10^10

36 7909306976 10 30 36

35 3954653488 10 29 35

34 1977326744 10 28 34

33 1129901000 10 28 33

32 564950500 9 27 32

31 282475250 9 26 31

30 161414432 9 25 30

29 80707216 8 24 29

28 40353608 8 23 28

27 23059208 8 23 27

26 11529604 8 22 26

25 5764802 7 21 25

24 3294176 7 20 24

23 1647088 7 19 23

22 823544 6 18 22

21 470600 6 18 21

20 235300 6 17 20

19 117650 6 16 19

18 67232 5 15 18

17 33616 5 14 17

16 16808 5 13 16

15 9608 4 12 15

14 4804 4 12 14

13 2402 4 11 13

12 1376 4 10 12

11 688 3 9 11

10 344 3 8 10

9 200 3 7 9

8 100 3 7 8

7 50 2 6 7

6 16 2 4 5

5 8 1 3 4

4 4 1 2 3

3 Prime!    2 1 1 2

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 8 is a palindrome
A046239 | A046240
L root 10
L cube 10
L base 8

(Recalc) Next > 10^10

21 68719738881 11 33 37

20 68719476737 11 33 37

19 8589934593 10 30 34

18 1073774593 10 28 31

17 1073741825 10 28 31

16 134217729 9 25 28

15 16781313 8 22 25

14 16777217 8 22 25

13 2097153 7 19 22

12 Prime!    262657 6 17 19

11 262145 6 17 19

10 32769 5 14 16

9 4161 4 11 13

8 4097 4 11 13

7 513 3 9 10

6 Prime!    73 2 6 7

5 65 2 6 7

4 9 1 3 4

3 Prime!    3 1 2 2

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 9 is a palindrome
A046241 | A046242
L root 10
L cube 10
L base 9

(Recalc) Next > 10^10

20 31381059610 11 32 34

19 3486843451 10 29 31

18 3486784402 10 29 31

17 387420490 9 26 28

16 43053283 8 23 25

15 43046722 8 23 25

14 4782970 7 21 22

13 532171 6 18 19

12 531442 6 18 19

11 59050 5 15 16

10 6643 4 12 13

9 6562 4 12 13

8 730 3 9 10

7 91 2 6 7

6 82 2 6 7

5 38 2 5 5

4 10 2 4 4

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 11 is a palindrome
A046243 | A046244
L root 10
L cube 10
L base 11

Next > 10^10

30 2572306584 10 29 28

29 2377434984 10 29 28

28 2358123384 10 29 28

27 2357947692 10 29 28

26 233846064 9 26 25

25 216130564 9 26 25

24 214521264 9 25 25

23 214373523 9 25 25

22 214358882 9 25 25

21 21258744 8 22 22

20 19648344 8 22 22

19 19503144 8 22 22

18 19487172 8 22 22

17 1932624 7 19 19

16 Prime!    1772893 7 19 19

15 1771562 7 19 19

14 175704 6 16 16

13 162504 6 16 16

12 161052 6 16 16

11 15984 5 13 13

10 14763 5 13 13

9 14642 5 13 13

8 1332 4 10 10

7 133 3 7 7

6 122 3 7 7

5 12 2 4 4

4 Prime!    7 1 3 3

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 12 is a palindrome
A046245 | A046246
L root 10
L cube 10
L base 12

Next > 10^10

29 5589762061 10 30 28

28 5195612305 10 30 28

27 5160049921 10 30 28

26 5159780353 10 30 28

25 465813517 9 27 25

24 432967825 9 26 25

23 430232257 9 26 25

22 430002433 9 26 25

21 429981697 9 26 25

20 38817805 8 23 22

19 36080785 8 23 22

18 35854273 8 23 22

17 35831809 8 23 22

16 3234829 7 20 19

15 2987713 7 20 19

14 2985985 7 20 19

13 269581 6 17 16

12 250705 6 17 16

11 248833 6 17 16

10 22477 5 14 13

9 20881 5 13 13

8 20737 5 13 13

7 1729 4 10 10

6 Prime!    157 3 7 7

5 145 3 7 7

4 Prime!    13 2 4 4

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 13 is a palindrome
A046247 | A046248
L root 10
L cube 10
L base 13

Next > 10^10

27 878479252 9 27 25

26 820557700 9 27 25

25 816104212 9 27 25

24 815759283 9 27 25

23 815730722 9 27 25

22 67575340 8 24 22

21 63119980 8 24 22

20 62779276 8 24 22

19 62748518 8 24 22

18 5198116 7 21 19

17 4855540 7 21 19

16 4829007 7 21 19

15 4826810 7 21 19

14 399868 6 17 16

13 373660 6 17 16

12 371294 6 17 16

11 30772 5 14 13

10 28731 5 14 13

9 28562 5 14 13

8 2380 4 11 10

7 2198 4 11 10

6 183 3 7 7

5 170 3 7 7

4 14 2 4 4

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 14 is a palindrome
A046249 | A046250
L root 10
L cube 10
L base 14

Next > 10^10

31 1581240991 10 28 25

30 1581202575 10 28 25

29 1483318789 10 28 25

28 Prime!    1476368041 10 28 25

27 1476329625 10 28 25

26 1475827473 10 28 25

25 1475789057 10 28 25

24 112943055 9 25 22

23 105951525 9 25 22

22 105454665 9 25 22

21 105413505 9 25 22

20 Prime!    8070119 7 21 19

19 8067375 7 21 19

18 7568149 7 21 19

17 7532281 7 21 19

16 7529537 7 21 19

15 576255 6 18 16

14 540765 6 18 16

13 537825 6 18 16

12 41175 5 14 13

11 38613 5 14 13

10 38417 5 14 13

9 2955 4 11 10

8 2745 4 11 10

7 Prime!    461 3 8 7

6 Prime!    211 3 7 7

5 Prime!    197 3 7 7

4 15 2 4 4

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 15 is a palindrome
A046251 | A046252
L root 10
L cube 10
L base 15

Next > 10^10

44 2733800641 10 29 25

43 2733750016 10 29 25

42 2574281476 10 29 25

41 Prime!    2563704001 10 29 25

40 2563653376 10 29 25

39 2562941251 10 29 25

38 2562890626 10 29 25

37 1366875008 10 28 24

36 683437504 9 27 23

35 182250016 9 25 22

34 171618976 9 25 22

33 170913376 9 25 22

32 170859376 9 25 22

31 91125008 8 24 21

30 45562504 8 23 20

29 12153391 8 22 19

28 12150016 8 22 19

27 11441476 8 22 19

26 11394001 8 22 19

25 11390626 8 22 19

24 6075008 7 21 18

23 3037504 7 20 17

22 810016 6 18 16

21 762976 6 18 16

20 759376 6 18 16

19 405008 6 17 15

18 202504 6 16 14

17 54016 5 15 13

16 50851 5 15 13

15 50626 5 15 13

14 27008 5 14 12

13 13504 5 13 11

12 3616 4 11 10

11 3376 4 11 10

10 1808 4 10 9

9 904 3 9 8

8 Prime!    241 3 8 7

7 226 3 8 7

6 16 2 4 4

5 8 1 3 3

4 4 1 2 2

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 16 is a palindrome
A029735 | A029736
L root 10
L cube 10
L base 16

(Recalc) Next > 10^10

49 1168499544081 13 37 31

48 1168247947281 13 37 31

47 1168232153105 13 37 31

46 1168231104529 13 37 31

45 1104075034881 13 37 31

44 1103807643905 13 37 31

43 1103806595329 13 37 31

42 1099781115905 13 37 31

41 1099780067329 13 37 31

40 1099529519105 13 37 31

39 1099528470529 13 37 31

38 1099512676353 13 37 31

37 1099511627777 13 37 31

36 73014444049 11 33 28

35 68987912449 11 33 28

34 68736258049 11 33 28

33 68720590849 11 33 28

32 68719476737 11 33 28

31 4563468305 10 29 25

30 4563402769 10 29 25

29 4311744769 10 29 25

28 4296085505 10 29 25

27 4296019969 10 29 25

26 4295032833 10 29 25

25 4294967297 10 29 25

24 285212689 9 26 22

23 269484289 9 26 22

22 268505089 9 26 22

21 268435457 9 26 22

20 17829905 8 22 19

19 17825809 8 22 19

18 16847105 8 22 19

17 16843009 8 22 19

16 16781313 8 22 19

15 16777217 8 22 19

14 1114129 7 19 16

13 1052929 7 19 16

12 1048577 7 19 16

11 69649 5 15 13

10 65793 5 15 13

9 Prime!    65537 5 15 13

8 4369 4 11 10

7 4097 4 11 10

6 273 3 8 7

5 Prime!    257 3 8 7

4 Prime!    17 2 4 4

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

Index Nr
Decimal equivalent of numbers whose
cube in base 17 is a palindrome
L root 10
L cube 10
L base 17

Next > 10^10

33 7386179653 10 30 25

32 7386096132 10 30 25

31 6999895300 10 30 25

30 6977265733 10 30 25

29 6977182212 10 30 25

28 6975840963 10 30 25

27 6975757442 10 30 25

26 434476260 9 26 22

25 411758820 9 26 22

24 410427108 9 26 22

23 410338674 9 26 22

22 25562357 8 23 19

21 25557444 8 23 19

20 24226293 8 23 19

19 24221380 8 23 19

18 24142483 8 23 19

17 24137570 8 23 19

16 1503396 7 19 16

15 1425060 7 19 16

14 1419858 7 19 16

13 88452 5 15 13

12 83811 5 15 13

11 83522 5 15 13

10 5220 4 12 10

9 4914 4 12 10

8 3377 4 11 9

7 Prime!    307 3 8 7

6 290 3 8 7

5 18 2 4 4

4 6 1 3 2

3 Prime!    2 1 1 1

2 1 1 1 1

1 0 1 1 1

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

1048577³ = 1152924803144876033 = 1000030000300001_{16}

You added

1152921504606846977³ = 1532495540865888862346031014505056805788924816845176833 =

1000000000000003000000000000003000000000000001_{16}

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 :

['1', '0', '4', '0', '6', '0', '4', '0', '1']

and on and on.

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.

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Patrick De Geest - Belgium

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E-mail address : pdg@worldofnumbers.com