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Palindromic Cubes
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Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
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

( base ) x ( base ) x ( base )orbase 3

bu17 The best way to get a 'structural' insight and how to imagine cubes is to visit these sites :

bu17 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, ...

cubespin BaseCube
111331
1011030301
10011003003001
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 :

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.

[480] 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.

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

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.

Walter Schneider (email) searched up to 18-digit basenumbers inclusive [ November 13, 2002 ].

Index NrInfo BasenumberLength
Palindromic Cube Number [Formula n^3]Length
cube cube
 
88 110.000.000.000.000.01118
1.331.000.000.000.000.399.300.000.000.000.039.930.000.000.000.001.33152
87 101.000.000.000.000.10118
1.030.301.000.000.003.090.903.000.000.003.090.903.000.000.001.030.30152
86 100.100.000.000.001.00118
1.003.003.001.000.030.090.090.030.000.300.900.900.300.001.003.003.00152
85 100.010.000.000.010.00118
1.000.300.030.001.300.090.009.000.330.009.000.900.031.000.300.030.00152
84 100.001.000.000.100.00118
1.000.030.000.303.001.090.000.903.003.090.000.901.003.030.000.300.00152
83 100.000.100.001.000.00118
1.000.003.000.033.000.091.000.390.000.930.001.900.003.300.003.000.00152
82 100.000.010.010.000.00118
1.000.000.300.300.030.090.031.009.009.001.300.900.300.030.030.000.00152
81 100.000.001.100.000.00118
1.000.000.033.000.000.393.000.001.991.000.003.930.000.003.300.000.00152
80 100.000.000.000.000.00118
1.000.000.000.000.000.030.000.000.000.000.000.300.000.000.000.000.00152
79 11.000.000.000.000.01117
1.331.000.000.000.003.993.000.000.000.003.993.000.000.000.001.33149
78 10.100.000.000.000.10117
1.030.301.000.000.030.909.030.000.000.309.090.300.000.001.030.30149
77 10.010.000.000.001.00117
1.003.003.001.000.300.900.900.300.030.090.090.030.001.003.003.00149
76 10.001.000.000.010.00117
1.000.300.030.004.000.900.090.006.000.900.090.004.000.300.030.00149
75 10.000.100.000.100.00117
1.000.030.000.330.001.900.009.300.039.000.091.000.330.000.300.00149
74 10.000.010.001.000.00117
1.000.003.000.303.000.901.030.900.090.301.090.003.030.003.000.00149
73 10.000.001.010.000.00117
1.000.000.303.000.030.903.001.090.901.003.090.300.003.030.000.00149
72 10.000.000.100.000.00117
1.000.000.030.000.000.600.000.007.000.000.060.000.000.300.000.00149
71 10.000.000.000.000.00117
1.000.000.000.000.000.300.000.000.000.000.030.000.000.000.000.00149
70 1.100.000.000.000.01116
1.331.000.000.000.039.930.000.000.000.399.300.000.000.001.33146
69 1.010.000.000.000.10116
1.030.301.000.000.309.090.300.000.030.909.030.000.001.030.30146
68 1.001.000.000.001.00116
1.003.003.001.003.009.009.003.003.009.009.003.001.003.003.00146
67 1.000.100.000.010.00116
1.000.300.030.031.009.000.900.330.090.009.001.300.300.030.00146
66 1.000.001.001.000.00116
1.000.003.003.003.009.004.009.009.004.009.003.003.003.000.00146
65 1.000.000.110.000.00116
1.000.000.330.000.039.300.001.991.000.039.300.000.330.000.00146
64 1.000.000.000.000.00116
1.000.000.000.000.003.000.000.000.000.003.000.000.000.000.00146
63 110.000.000.000.01115
1.331.000.000.000.399.300.000.000.039.930.000.000.001.33143
62 101.000.000.000.10115
1.030.301.000.003.090.903.000.003.090.903.000.001.030.30143
61 100.100.000.001.00115
1.003.003.001.030.090.090.030.300.900.900.301.003.003.00143
60 100.010.000.010.00115
1.000.300.030.301.090.009.030.309.000.901.030.300.030.00143
59 100.001.000.100.00115
1.000.030.003.300.091.003.900.093.001.900.033.000.300.00143
58 100.000.101.000.00115
1.000.003.030.003.090.301.090.901.030.903.000.303.000.00143
57 100.000.010.000.00115
1.000.000.300.000.060.000.007.000.000.600.000.030.000.00143
56 100.000.000.000.00115
1.000.000.000.000.030.000.000.000.000.300.000.000.000.00143
55 11.000.000.000.01114
1.331.000.000.003.993.000.000.003.993.000.000.001.33140
54 10.100.000.000.10114
1.030.301.000.030.909.030.000.309.090.300.001.030.30140
53 10.010.000.001.00114
1.003.003.001.300.900.900.330.090.090.031.003.003.00140
52 10.001.000.010.00114
1.000.300.033.001.900.093.003.900.091.003.300.030.00140
51 10.000.100.100.00114
1.000.030.030.300.901.309.009.031.090.030.300.300.00140
50 10.000.011.000.00114
1.000.003.300.003.930.001.991.000.393.000.033.000.00140
49 10.000.000.000.00114
1.000.000.000.000.300.000.000.000.030.000.000.000.00140
48 1.100.000.000.01113
1.331.000.000.039.930.000.000.399.300.000.001.33137
47 1.010.000.000.10113
1.030.301.000.309.090.300.030.909.030.001.030.30137
46 1.001.000.001.00113
1.003.003.004.009.009.006.009.009.004.003.003.00137
45 1.000.010.100.00113
1.000.030.300.309.031.090.901.309.030.030.300.00137
44 1.000.001.000.00113
1.000.003.000.006.000.007.000.006.000.003.000.00137
43 1.000.000.000.00113
1.000.000.000.003.000.000.000.003.000.000.000.00137
42 110.000.000.01112
1.331.000.000.399.300.000.039.930.000.001.33134
41 101.000.000.10112
1.030.301.003.090.903.003.090.903.001.030.30134
40 100.100.001.00112
1.003.003.031.090.090.330.900.901.303.003.00134
39 100.010.010.00112
1.000.300.330.091.039.009.301.900.330.030.00134
38 100.001.100.00112
1.000.033.000.393.001.991.003.930.003.300.00134
37 100.000.000.00112
1.000.000.000.030.000.000.000.300.000.000.00134
36 11.000.000.01111
1.331.000.003.993.000.003.993.000.001.33131
35 10.100.000.10111
1.030.301.030.909.030.309.090.301.030.30131
34 10.010.001.00111
1.003.003.301.900.930.390.091.033.003.00131
33 10.001.010.00111
1.000.303.030.904.090.904.090.303.030.00131
32 10.000.100.00111
1.000.030.000.600.007.000.060.000.300.00131
31 10.000.000.00111
1.000.000.000.300.000.000.030.000.000.00131
30 1.100.000.01110
1.331.000.039.930.000.399.300.001.33128
29 1.010.000.10110
1.030.301.309.090.330.909.031.030.30128
28 1.000.110.00110
1.000.330.039.301.991.039.300.330.00128
27 1.000.000.00110
1.000.000.003.000.000.003.000.000.00128
26 110.000.0119
1.331.000.399.300.039.930.001.33125
25 101.000.1019
1.030.304.090.906.090.904.030.30125
24 100.101.0019
1.003.033.091.390.931.903.303.00125
23 100.010.0019
1.000.300.060.007.000.600.030.00125
22 100.000.0019
1.000.000.030.000.000.300.000.00125
21 11.000.0118
1.331.003.993.003.993.001.33122
20 10.100.1018
1.030.331.909.339.091.330.30122
19 10.011.0018
1.003.303.931.991.393.033.00122
18 10.000.0018
1.000.000.300.000.030.000.00122
17 1.100.0117
1.331.039.930.399.301.33119
16 1.001.0017
1.003.006.007.006.003.00119
15 1.000.0017
1.000.003.000.003.000.00119
14 110.0116
1.331.399.339.931.33116
13 101.1016
1.033.394.994.933.30116
12 100.0016
1.000.030.000.300.00116
11 11.0115
1.334.996.994.33113
10 10.1015
1.030.607.060.30113
9 10.0015
1.000.300.030.00113
8 Info 2.2014
10.662.526.60111
7 1.0014
1.003.003.00110
6 1113
1.367.6317
5 1013
1.030.3017
4 112
1.3314
3 71
3433
2 21
81
1 11
11



Contributions

[ September 14, 2000 ] Walter Schneider (email)
searched up to 13 * 109 and found some
additional solutions. See Index Nrs 31 to 36.
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 37 to 42.
     
    [ 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 43 to 55.
    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 56-70);...
     
    [ November 13, 2002 ] Walter Schneider wrote :
    Searched completely for palindromic cubes with 17 and 18 digit
    basenumbers. Here are the results (indexes 71-88).
    Running time on a 2,4 GHz Pentium-4:
    11 hours for 17 digits and about 5 days for 18 digits.
    brown line
    triangle tetra square cube circular palpri
    brown line

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