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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 : 8, 3113, 44611644

Cubic numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

 or ( base ) x ( base ) x ( base ) base 3

The best way to get a 'structural' insight and how to imagine cubes is to visit these sites :

• From Eric Weisstein's Math Encyclopedia : Cubic Number

Some among us played around with cubes as the following sequences from Sloane's table testify :
%N Not the sum of distinct cubes. under A001476
%N Not the sum of 2 nonnegative cubes. under A022555
%N Not the sum of 3 nonnegative cubes. under A022561
%N Not the sum of 4 nonnegative cubes. under A022566
%N Not the sum of a cube and a triangular number. under A014156
%N Not the sum of 2 cubes and a triangular number. under A014158
%N Not the sum of a square and a nonnegative cube. under A022550
%N Not the sum of a square and 2 nonnegative cubes. under A022557
%N Not the sum of 2 squares and a nonnegative cube. under A022552

Palindromic Cubes

Unlike Palindromic Triangulars where it is impossible to predict a next higher one, whether its basenumber is palindromic or not,
with the Palindromic Cubes (and Squares) we have an opposite situation. Finding a next higher number is very easy. Start e.g. with the basenumber 11.
Then repeatedly add a zero between the two 'ones' and cube them.
A pattern emerges that can go on forever.

All cubic numbers can only end with digit... well  any digit !

Here is a synopsis of my humble 'palindromic cube' pattern-investigation
Four infinite patterns :
103n+3*102n+3*10n+1 » n = 1, 2, 3, 4, ... » 1331, 1030301, 1003003001, ...

Base Cube 11 1331 101 1030301 1001 1003003001 10001 1000300030001

106n+3*105n+6*104n+7*103n+6*102n+3*10n+1 » n = 1, 2, 3, ...
» 1367631, 1030607060301, 1003006007006003001, ...

C*103n+3C*102n+3C*10n+C » n = 3, 4, 5, 6, ... C = 1331 !
» 1334996994331, 1331399339931331, 1331039930399301331, 1331003993003993001331, ..
Note that when expanding with n = 3 we get an overlapping of numbers, but due to absence of any carry
we keep the palindromicy of the result :
```1331000000000
3993000000
3993000
1331 +
-------------
1334996994331
```

106n+3+33*105n+2+393*104n+1+1991*103n+393*102n+33*10n+1 » n = 2, 3, 4, 5, ...
» 1033394994933301, 1003303931991393033001, 1000330039301991039300330001, ...
Note that just like the previous one all the coefficients are palindromic !
Intricacy and beauty can go hand in hand sometimes.

For the moment only one nonpalindromic basenumber of a palindromic cube is known to me. It's 2201.
By the way, there exist a basenumber of a palindromic square (!) starting with that number.

[481] 2201019508986478

Some palindromic cubes can be expressed as the sum of two or more consecutive primes

8 = 3 + 5
1331 = 439 + 443 + 449

Sources Revealed

Here is extra bookinfo for the interested reader :

Here are Two Inquisitive Problems chosen from B.S. Rangaswamy's work “Wonders of Numerals”
[ISBN: 81-7478-492-6]. The first problem deals with palindromic cubes.

I found this in Martin Gardner's book “The Ambidextrous Universe” Second Edition (1982) page 40 :
Palindromic cubes whose cube root (i.e. basenumber) are not palindromes are so rare
that only one is known: 10,662,526,601 = 2,2013. No palindrome is known
that is an n_th power, n greater than 3, whose n_th root is not palindromic.

This info from Martin Gardner's book “Mathematical Circus” page 245
Cubes too are unusually rich in palindromes. A computer check on all cubes less than
2,8 x 1014 turned up a truly astonishing fact. The only palindromic cube with a
nonpalindromic cube root, among the cubes examined by Simmons, is 10,662,526,601.
Its cube root, 2,201, had been noticed earlier by Trigg, who reported in 1961 that
it was the only nonpalindrome with a palindromic cube less than 1,953,125,000,000.
It is not yet known if  2,201 is the only integer with this property.
Simmons' computer search of palindromic fourth powers, to the same limit
as his search of cubes, failed to uncover a single palindromic fourth power whose
fourth root was not a palindrome of the general form 10...01.
For powers 5 through 10 the computer found no palindromes at all except for the
trivial case of 1. Simmons conjectures that there are no palindromes of the form X k
where k is greater than 4.

Again another book from Martin Gardner is “Puzzles from other Worlds” page 107
The only known asymmetric number that produces a palindrome when cubed is  2201 .
Its cube is 10662526601. According to VOZ, this was first noted by Trigg in 1961.

Similar facts in David Wells' book “Curious and Interesting Numbers” page 193
The only known palindromic cube, 22013, whose root is not palindromic.
All known palindromic 4th powers have palindromic roots.
No palindromic 5th powers are known.

That was way back in 1988. Can't we do better, folks ...
Don't hesitate and send me more of such numbers (palindromic powers).
I'll be glad to include them in (the) tables.

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular cubes are categorised as follows : %N The cubes: a(n) = n^3. under A000578 The palindromic cubes are already categorised : %N Cube is a palindrome. under A002780 %N Palindromic cubes. under A002781 %N n and n^3 are both palindromes. under A069748 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.

The Table

Index NrInfoBasenumberLength
Palindromic Cube NumberLength

Formula n^3
89110.000.000.000.000.01118
1.331.000.000.000.000.399.300.000.000.000.039.930.000.000.000.001.33152
88101.000.000.000.000.10118
1.030.301.000.000.003.090.903.000.000.003.090.903.000.000.001.030.30152
87100.100.000.000.001.00118
1.003.003.001.000.030.090.090.030.000.300.900.900.300.001.003.003.00152
86100.010.000.000.010.00118
1.000.300.030.001.300.090.009.000.330.009.000.900.031.000.300.030.00152
85100.001.000.000.100.00118
1.000.030.000.303.001.090.000.903.003.090.000.901.003.030.000.300.00152
84100.000.100.001.000.00118
1.000.003.000.033.000.091.000.390.000.930.001.900.003.300.003.000.00152
83100.000.010.010.000.00118
1.000.000.300.300.030.090.031.009.009.001.300.900.300.030.030.000.00152
82100.000.001.100.000.00118
1.000.000.033.000.000.393.000.001.991.000.003.930.000.003.300.000.00152
81100.000.000.000.000.00118
1.000.000.000.000.000.030.000.000.000.000.000.300.000.000.000.000.00152
8011.000.000.000.000.01117
1.331.000.000.000.003.993.000.000.000.003.993.000.000.000.001.33149
7910.100.000.000.000.10117
1.030.301.000.000.030.909.030.000.000.309.090.300.000.001.030.30149
7810.010.000.000.001.00117
1.003.003.001.000.300.900.900.300.030.090.090.030.001.003.003.00149
7710.001.000.000.010.00117
1.000.300.030.004.000.900.090.006.000.900.090.004.000.300.030.00149
7610.000.100.000.100.00117
1.000.030.000.330.001.900.009.300.039.000.091.000.330.000.300.00149
7510.000.010.001.000.00117
1.000.003.000.303.000.901.030.900.090.301.090.003.030.003.000.00149
7410.000.001.010.000.00117
1.000.000.303.000.030.903.001.090.901.003.090.300.003.030.000.00149
7310.000.000.100.000.00117
1.000.000.030.000.000.600.000.007.000.000.060.000.000.300.000.00149
7210.000.000.000.000.00117
1.000.000.000.000.000.300.000.000.000.000.030.000.000.000.000.00149
711.100.000.000.000.01116
1.331.000.000.000.039.930.000.000.000.399.300.000.000.001.33146
701.010.000.000.000.10116
1.030.301.000.000.309.090.300.000.030.909.030.000.001.030.30146
691.001.000.000.001.00116
1.003.003.001.003.009.009.003.003.009.009.003.001.003.003.00146
681.000.100.000.010.00116
1.000.300.030.031.009.000.900.330.090.009.001.300.300.030.00146
671.000.001.001.000.00116
1.000.003.003.003.009.004.009.009.004.009.003.003.003.000.00146
661.000.000.110.000.00116
1.000.000.330.000.039.300.001.991.000.039.300.000.330.000.00146
651.000.000.000.000.00116
1.000.000.000.000.003.000.000.000.000.003.000.000.000.000.00146
64110.000.000.000.01115
1.331.000.000.000.399.300.000.000.039.930.000.000.001.33143
63101.000.000.000.10115
1.030.301.000.003.090.903.000.003.090.903.000.001.030.30143
62100.100.000.001.00115
1.003.003.001.030.090.090.030.300.900.900.301.003.003.00143
61100.010.000.010.00115
1.000.300.030.301.090.009.030.309.000.901.030.300.030.00143
60100.001.000.100.00115
1.000.030.003.300.091.003.900.093.001.900.033.000.300.00143
59100.000.101.000.00115
1.000.003.030.003.090.301.090.901.030.903.000.303.000.00143
58100.000.010.000.00115
1.000.000.300.000.060.000.007.000.000.600.000.030.000.00143
57100.000.000.000.00115
1.000.000.000.000.030.000.000.000.000.300.000.000.000.00143
5611.000.000.000.01114
1.331.000.000.003.993.000.000.003.993.000.000.001.33140
5510.100.000.000.10114
1.030.301.000.030.909.030.000.309.090.300.001.030.30140
5410.010.000.001.00114
1.003.003.001.300.900.900.330.090.090.031.003.003.00140
5310.001.000.010.00114
1.000.300.033.001.900.093.003.900.091.003.300.030.00140
5210.000.100.100.00114
1.000.030.030.300.901.309.009.031.090.030.300.300.00140
5110.000.011.000.00114
1.000.003.300.003.930.001.991.000.393.000.033.000.00140
5010.000.000.000.00114
1.000.000.000.000.300.000.000.000.030.000.000.000.00140
491.100.000.000.01113
1.331.000.000.039.930.000.000.399.300.000.001.33137
481.010.000.000.10113
1.030.301.000.309.090.300.030.909.030.001.030.30137
471.001.000.001.00113
1.003.003.004.009.009.006.009.009.004.003.003.00137
461.000.010.100.00113
1.000.030.300.309.031.090.901.309.030.030.300.00137
451.000.001.000.00113
1.000.003.000.006.000.007.000.006.000.003.000.00137
441.000.000.000.00113
1.000.000.000.003.000.000.000.003.000.000.000.00137
43110.000.000.01112
1.331.000.000.399.300.000.039.930.000.001.33134
42101.000.000.10112
1.030.301.003.090.903.003.090.903.001.030.30134
41100.100.001.00112
1.003.003.031.090.090.330.900.901.303.003.00134
40100.010.010.00112
1.000.300.330.091.039.009.301.900.330.030.00134
39100.001.100.00112
1.000.033.000.393.001.991.003.930.003.300.00134
38100.000.000.00112
1.000.000.000.030.000.000.000.300.000.000.00134
3711.000.000.01111
1.331.000.003.993.000.003.993.000.001.33131
3610.100.000.10111
1.030.301.030.909.030.309.090.301.030.30131
3510.010.001.00111
1.003.003.301.900.930.390.091.033.003.00131
3410.001.010.00111
1.000.303.030.904.090.904.090.303.030.00131
3310.000.100.00111
1.000.030.000.600.007.000.060.000.300.00131
3210.000.000.00111
1.000.000.000.300.000.000.030.000.000.00131
311.100.000.01110
1.331.000.039.930.000.399.300.001.33128
301.010.000.10110
1.030.301.309.090.330.909.031.030.30128
291.000.110.00110
1.000.330.039.301.991.039.300.330.00128
281.000.000.00110
1.000.000.003.000.000.003.000.000.00128
27110.000.0119
1.331.000.399.300.039.930.001.33125
26101.000.1019
1.030.304.090.906.090.904.030.30125
25100.101.0019
1.003.033.091.390.931.903.303.00125
24100.010.0019
1.000.300.060.007.000.600.030.00125
23100.000.0019
1.000.000.030.000.000.300.000.00125
2211.000.0118
1.331.003.993.003.993.001.33122
2110.100.1018
1.030.331.909.339.091.330.30122
2010.011.0018
1.003.303.931.991.393.033.00122
1910.000.0018
1.000.000.300.000.030.000.00122
181.100.0117
1.331.039.930.399.301.33119
171.001.0017
1.003.006.007.006.003.00119
161.000.0017
1.000.003.000.003.000.00119
15110.0116
1.331.399.339.931.33116
14101.1016
1.033.394.994.933.30116
13100.0016
1.000.030.000.300.00116
1211.0115
1.334.996.994.33113
1110.1015
1.030.607.060.30113
1010.0015
1.000.300.030.00113
9Info2.2014
10.662.526.60111
81.0014
1.003.003.00110
71113
1.367.6317
61013
1.030.3017
5112
1.3314
471
3433
321
81
211
11
101
01

Contributions

[ September 14, 2000 ] Walter Schneider (email)
searched up to 13 * 109 and found some
additional solutions. See Index Nrs [32] to [37].
The running time was about 3 hours for checking 109 numbers.

[ October 3, 2000 ] Walter Schneider wrote :
I have speeded up my program for searching palindromic cubes
using the fact that for (a.10n + b)3
• the first digits are the same as a3 and
• the last digits are the same as b3.
• Total running time was 45 minutes for checking up to the 12-digit basenumbers.
See Index Nrs [38] to [43].

[ October 5, 2000 ] Walter Schneider wrote :
I have completed the search of palindromic cubes for 13- and 14-digit basenumbers.
As you see there are only the known 0/1-solutions. See Index Nrs [44] to [56].
At the moment I have stopped the search because the limits are reached.
The last search for the 14-digit basenumbers has needed 9 hours and searching
15-digit basenumbers will therefore need some days of computing time.

[ May 24, 2002 ] Walter Schneider wrote :
I searched completely for 15 and 16 digit basenumbers.
Here are the results (indexes [57] up to [71])

[ November 13, 2002 ] Walter Schneider (email) wrote :
Searched completely for palindromic cubes with 17 and 18 digit
basenumbers. Here are the results (indexes [72] up to [89]).
Running time on a 2,4 GHz Pentium-4:
11 hours for 17 digits and about 5 days for 18 digits.

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