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Palindromic Cubes
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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

( base ) x ( base ) x ( base )	or	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
• Secondly read this general introduction about Cubes by Richard Phillips.

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

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.

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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

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

8 = 3 + 5

1331 = 439 + 443 + 449

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 4^th powers have palindromic roots.

No palindromic 5^th 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.

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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.

Index Nr	Info	Basenumber	Length
	Palindromic Cube Number		Length
	Formula n^3
89	110.000.000.000.000.011		18
	1.331.000.000.000.000.399.300.000.000.000.039.930.000.000.000.001.331		52
88	101.000.000.000.000.101		18
	1.030.301.000.000.003.090.903.000.000.003.090.903.000.000.001.030.301		52
87	100.100.000.000.001.001		18
	1.003.003.001.000.030.090.090.030.000.300.900.900.300.001.003.003.001		52
86	100.010.000.000.010.001		18
	1.000.300.030.001.300.090.009.000.330.009.000.900.031.000.300.030.001		52
85	100.001.000.000.100.001		18
	1.000.030.000.303.001.090.000.903.003.090.000.901.003.030.000.300.001		52
84	100.000.100.001.000.001		18
	1.000.003.000.033.000.091.000.390.000.930.001.900.003.300.003.000.001		52
83	100.000.010.010.000.001		18
	1.000.000.300.300.030.090.031.009.009.001.300.900.300.030.030.000.001		52
82	100.000.001.100.000.001		18
	1.000.000.033.000.000.393.000.001.991.000.003.930.000.003.300.000.001		52
81	100.000.000.000.000.001		18
	1.000.000.000.000.000.030.000.000.000.000.000.300.000.000.000.000.001		52
80	11.000.000.000.000.011		17
	1.331.000.000.000.003.993.000.000.000.003.993.000.000.000.001.331		49
79	10.100.000.000.000.101		17
	1.030.301.000.000.030.909.030.000.000.309.090.300.000.001.030.301		49
78	10.010.000.000.001.001		17
	1.003.003.001.000.300.900.900.300.030.090.090.030.001.003.003.001		49
77	10.001.000.000.010.001		17
	1.000.300.030.004.000.900.090.006.000.900.090.004.000.300.030.001		49
76	10.000.100.000.100.001		17
	1.000.030.000.330.001.900.009.300.039.000.091.000.330.000.300.001		49
75	10.000.010.001.000.001		17
	1.000.003.000.303.000.901.030.900.090.301.090.003.030.003.000.001		49
74	10.000.001.010.000.001		17
	1.000.000.303.000.030.903.001.090.901.003.090.300.003.030.000.001		49
73	10.000.000.100.000.001		17
	1.000.000.030.000.000.600.000.007.000.000.060.000.000.300.000.001		49
72	10.000.000.000.000.001		17
	1.000.000.000.000.000.300.000.000.000.000.030.000.000.000.000.001		49
71	1.100.000.000.000.011		16
	1.331.000.000.000.039.930.000.000.000.399.300.000.000.001.331		46
70	1.010.000.000.000.101		16
	1.030.301.000.000.309.090.300.000.030.909.030.000.001.030.301		46
69	1.001.000.000.001.001		16
	1.003.003.001.003.009.009.003.003.009.009.003.001.003.003.001		46
68	1.000.100.000.010.001		16
	1.000.300.030.031.009.000.900.330.090.009.001.300.300.030.001		46
67	1.000.001.001.000.001		16
	1.000.003.003.003.009.004.009.009.004.009.003.003.003.000.001		46
66	1.000.000.110.000.001		16
	1.000.000.330.000.039.300.001.991.000.039.300.000.330.000.001		46
65	1.000.000.000.000.001		16
	1.000.000.000.000.003.000.000.000.000.003.000.000.000.000.001		46
64	110.000.000.000.011		15
	1.331.000.000.000.399.300.000.000.039.930.000.000.001.331		43
63	101.000.000.000.101		15
	1.030.301.000.003.090.903.000.003.090.903.000.001.030.301		43
62	100.100.000.001.001		15
	1.003.003.001.030.090.090.030.300.900.900.301.003.003.001		43
61	100.010.000.010.001		15
	1.000.300.030.301.090.009.030.309.000.901.030.300.030.001		43
60	100.001.000.100.001		15
	1.000.030.003.300.091.003.900.093.001.900.033.000.300.001		43
59	100.000.101.000.001		15
	1.000.003.030.003.090.301.090.901.030.903.000.303.000.001		43
58	100.000.010.000.001		15
	1.000.000.300.000.060.000.007.000.000.600.000.030.000.001		43
57	100.000.000.000.001		15
	1.000.000.000.000.030.000.000.000.000.300.000.000.000.001		43
56	11.000.000.000.011		14
	1.331.000.000.003.993.000.000.003.993.000.000.001.331		40
55	10.100.000.000.101		14
	1.030.301.000.030.909.030.000.309.090.300.001.030.301		40
54	10.010.000.001.001		14
	1.003.003.001.300.900.900.330.090.090.031.003.003.001		40
53	10.001.000.010.001		14
	1.000.300.033.001.900.093.003.900.091.003.300.030.001		40
52	10.000.100.100.001		14
	1.000.030.030.300.901.309.009.031.090.030.300.300.001		40
51	10.000.011.000.001		14
	1.000.003.300.003.930.001.991.000.393.000.033.000.001		40
50	10.000.000.000.001		14
	1.000.000.000.000.300.000.000.000.030.000.000.000.001		40
49	1.100.000.000.011		13
	1.331.000.000.039.930.000.000.399.300.000.001.331		37
48	1.010.000.000.101		13
	1.030.301.000.309.090.300.030.909.030.001.030.301		37
47	1.001.000.001.001		13
	1.003.003.004.009.009.006.009.009.004.003.003.001		37
46	1.000.010.100.001		13
	1.000.030.300.309.031.090.901.309.030.030.300.001		37
45	1.000.001.000.001		13
	1.000.003.000.006.000.007.000.006.000.003.000.001		37
44	1.000.000.000.001		13
	1.000.000.000.003.000.000.000.003.000.000.000.001		37
43	110.000.000.011		12
	1.331.000.000.399.300.000.039.930.000.001.331		34
42	101.000.000.101		12
	1.030.301.003.090.903.003.090.903.001.030.301		34
41	100.100.001.001		12
	1.003.003.031.090.090.330.900.901.303.003.001		34
40	100.010.010.001		12
	1.000.300.330.091.039.009.301.900.330.030.001		34
39	100.001.100.001		12
	1.000.033.000.393.001.991.003.930.003.300.001		34
38	100.000.000.001		12
	1.000.000.000.030.000.000.000.300.000.000.001		34
37	11.000.000.011		11
	1.331.000.003.993.000.003.993.000.001.331		31
36	10.100.000.101		11
	1.030.301.030.909.030.309.090.301.030.301		31
35	10.010.001.001		11
	1.003.003.301.900.930.390.091.033.003.001		31
34	10.001.010.001		11
	1.000.303.030.904.090.904.090.303.030.001		31
33	10.000.100.001		11
	1.000.030.000.600.007.000.060.000.300.001		31
32	10.000.000.001		11
	1.000.000.000.300.000.000.030.000.000.001		31
31	1.100.000.011		10
	1.331.000.039.930.000.399.300.001.331		28
30	1.010.000.101		10
	1.030.301.309.090.330.909.031.030.301		28
29	1.000.110.001		10
	1.000.330.039.301.991.039.300.330.001		28
28	1.000.000.001		10
	1.000.000.003.000.000.003.000.000.001		28
27	110.000.011		9
	1.331.000.399.300.039.930.001.331		25
26	101.000.101		9
	1.030.304.090.906.090.904.030.301		25
25	100.101.001		9
	1.003.033.091.390.931.903.303.001		25
24	100.010.001		9
	1.000.300.060.007.000.600.030.001		25
23	100.000.001		9
	1.000.000.030.000.000.300.000.001		25
22	11.000.011		8
	1.331.003.993.003.993.001.331		22
21	10.100.101		8
	1.030.331.909.339.091.330.301		22
20	10.011.001		8
	1.003.303.931.991.393.033.001		22
19	10.000.001		8
	1.000.000.300.000.030.000.001		22
18	1.100.011		7
	1.331.039.930.399.301.331		19
17	1.001.001		7
	1.003.006.007.006.003.001		19
16	1.000.001		7
	1.000.003.000.003.000.001		19
15	110.011		6
	1.331.399.339.931.331		16
14	101.101		6
	1.033.394.994.933.301		16
13	100.001		6
	1.000.030.000.300.001		16
12	11.011		5
	1.334.996.994.331		13
11	10.101		5
	1.030.607.060.301		13
10	10.001		5
	1.000.300.030.001		13
9	Info	2.201	4
	10.662.526.601		11
8	1.001		4
	1.003.003.001		10
7	111		3
	1.367.631		7
6	101		3
	1.030.301		7
5	11		2
	1.331		4
4	7		1
	343		3
3	2		1
	8		1
2	1		1
	1		1
1	0		1
	0		1

[ 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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( © All rights reserved ) - Last modified : July 2, 2023.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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