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