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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

1. Go directly to the Palindromic cubes in bases 2 to 17 tables
2. Go directly to the Base 16 topic
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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
21111
10111

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
3Prime!    2112
21111
10111

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
184294967297102949
171073741825102846
1626843545792643
156710886582440
141677721782237
13419430572034
12104857771931
1126214561728
10Prime!    6553751525
91638551322
8409741119
7102541016
6Prime!    2573813
5652610
4Prime!    17247
3Prime!    5134
21111
10111

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
23476837158203126154564
2295367431640626144261
2119073486328126144058
203814697265626133855
19762939453126123652
18152587890626123449
1730517578126113246
166103515626103043
151220703126102840
1424414062692637
134882812682434
12976562672131
11195312671928
1039062661725
97812651522
81562651319
7312641116
66263913
51263710
426257
36134
21111
10111

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
152176782337102937
1486062745692735
1336279705792634
126046617782431
111007769782228
10167961771925
927993761722
84665751519
7777741216
6Prime!    129741013
52173810
4Prime!    37257
3Prime!    7134
21111
10111

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
367909306976103036
353954653488102935
341977326744102834
331129901000102833
3256495050092732
3128247525092631
3016141443292530
298070721682429
284035360882328
272305920882327
261152960482226
25576480272125
24329417672024
23164708871923
2282354461822
2147060061821
2023530061720
1911765061619
186723251518
173361651417
161680851316
15960841215
14480441214
13240241113
12137641012
116883911
103443810
9200379
8100378
750267
616245
58134
44123
3Prime!    2112
21111
10111

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
2168719738881113337
2068719476737113337
198589934593103034
181073774593102831
171073741825102831
1613421772992528
151678131382225
141677721782225
13209715371922
12Prime!    26265761719
1126214561719
103276951416
9416141113
8409741113
75133910
6Prime!    73267
565267
49134
3Prime!    3122
21111
10111

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
2031381059610113234
193486843451102931
183486784402102931
1738742049092628
164305328382325
154304672282325
14478297072122
1353217161819
1253144261819
115905051516
10664341213
9656241213
87303910
791267
682267
538255
410244
3Prime!    2111
21111
10111

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
302572306584102928
292377434984102928
282358123384102928
272357947692102928
2623384606492625
2521613056492625
2421452126492525
2321437352392525
2221435888292525
212125874482222
201964834482222
191950314482222
181948717282222
17193262471919
16Prime!    177289371919
15177156271919
1417570461616
1316250461616
1216105261616
111598451313
101476351313
91464251313
8133241010
7133377
6122377
512244
4Prime!    7133
3Prime!    2111
21111
10111

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
295589762061103028
285195612305103028
275160049921103028
265159780353103028
2546581351792725
2443296782592625
2343023225792625
2243000243392625
2142998169792625
203881780582322
193608078582322
183585427382322
173583180982322
16323482972019
15298771372019
14298598572019
1326958161716
1225070561716
1124883361716
102247751413
92088151313
82073751313
7172941010
6Prime!    157377
5145377
4Prime!    13244
3Prime!    2111
21111
10111

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
2787847925292725
2682055770092725
2581610421292725
2481575928392725
2381573072292725
226757534082422
216311998082422
206277927682422
196274851882422
18519811672119
17485554072119
16482900772119
15482681072119
1439986861716
1337366061716
1237129461716
113077251413
102873151413
92856251413
8238041110
7219841110
6183377
5170377
414244
3Prime!    2111
21111
10111

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
311581240991102825
301581202575102825
291483318789102825
28Prime!    1476368041102825
271476329625102825
261475827473102825
251475789057102825
2411294305592522
2310595152592522
2210545466592522
2110541350592522
20Prime!    807011972119
19806737572119
18756814972119
17753228172119
16752953772119
1557625561816
1454076561816
1353782561816
124117551413
113861351413
103841751413
9295541110
8274541110
7Prime!    461387
6Prime!    211377
5Prime!    197377
415244
3Prime!    2111
21111
10111

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
442733800641102925
432733750016102925
422574281476102925
41Prime!    2563704001102925
402563653376102925
392562941251102925
382562890626102925
371366875008102824
3668343750492723
3518225001692522
3417161897692522
3317091337692522
3217085937692522
319112500882421
304556250482320
291215339182219
281215001682219
271144147682219
261139400182219
251139062682219
24607500872118
23303750472017
2281001661816
2176297661816
2075937661816
1940500861715
1820250461614
175401651513
165085151513
155062651513
142700851412
131350451311
12361641110
11337641110
1018084109
9904398
8Prime!    241387
7226387
616244
58133
44122
3Prime!    2111
21111
10111

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
491168499544081133731
481168247947281133731
471168232153105133731
461168231104529133731
451104075034881133731
441103807643905133731
431103806595329133731
421099781115905133731
411099780067329133731
401099529519105133731
391099528470529133731
381099512676353133731
371099511627777133731
3673014444049113328
3568987912449113328
3468736258049113328
3368720590849113328
3268719476737113328
314563468305102925
304563402769102925
294311744769102925
284296085505102925
274296019969102925
264295032833102925
254294967297102925
2428521268992622
2326948428992622
2226850508992622
2126843545792622
201782990582219
191782580982219
181684710582219
171684300982219
161678131382219
151677721782219
14111412971916
13105292971916
12104857771916
116964951513
106579351513
9Prime!    6553751513
8436941110
7409741110
6273387
5Prime!    257387
4Prime!    17244
3Prime!    2111
21111
10111

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
337386179653103025
327386096132103025
316999895300103025
306977265733103025
296977182212103025
286975840963103025
276975757442103025
2643447626092622
2541175882092622
2441042710892622
2341033867492622
222556235782319
212555744482319
202422629382319
192422138082319
182414248382319
172413757082319
16150339671916
15142506071916
14141985871916
138845251513
128381151513
118352251513
10522041210
9491441210
833774119
7Prime!    307387
6290387
518244
46132
3Prime!    2111
21111
10111

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Base 16

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

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

Here is my reply after some perusing (PDG):

I found

10485773 = 1152924803144876033 = 1000030000300001{16}

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 :

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.

Contributions
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