Palindromic Mappings

Author: Robleh Wais 10/23/22

I have become interested in a topic that should be classified as part of number theory. This curiosity was sparked in me after observing the screen name of a youtuber which happens to be a palindromic sequence of letters: xylyxylyx.

After looking at them with intriguing interest for some time (I visit his YouTube site regularly), it occurred to me that this palindrome could be mapped to natural numbers (that is excluding zero). After that, the integers could be remapped to other palindromes via an operation like multiplication. From here, it was an obvious inference to suppose that an extension of the Palindromic Challenge could be created. Below is a description of the process. It is leading me to other perhaps more revealing areas to explore beyond what I am calling palindromic mappings.

The question is simple. But first we must construct the set.

If we take a N-tuple, where N=5, and use a seed S, where S=3, then we can generate an integral palindromic set for the 5-tuple, for S=(A,B,C) we have:

A

B

C

B

A

0 is excluded, so we use the whole number subset of the integers. For the first set, where s= (1,2,3), we have the 5-tuple string

1

2

3

2

1

If we multiply this 5-tuple string, the result is 12. If we set S= (4,5,6), then we have the 5-tuple string 4,5,6,5,4, If we multiple this 5-tuple string, the result is 2400.

 

Now, we can ask the question. Does any 5-tuple string, which uses increasing S subsets to generate palindromic sets, become a palindromic set? I've created an excel sheet that seeks to map these palindromic sets. But to do this, a few more operations have to be included. Please note this is related to, but not the same as, the Palindrome Problem:

Palindrome Conjecture

This problem says given an integer x= n (where n is number of digits) and xr is the reverse of x =n digits. F(x)= x+ xr . Check if F(x) is a palindrome. If it is not, iterate the process until a palindrome is the result.

Palindrome Multiplicative Mapping Conjecture

My problem is different. I want to know if a palindrome can produce another palindrome by mapping them with the operation * (multiplication).

So, to state the conjecture in formal language. We have the following:

Given the initial set S=1,2,3. S can generate a 5-tuple set Tn and it is palindromic, then for n=1 to 5 of T does the following construct generate another Tp

Multiplicative Palindrome Conjecture

 

Of course, if this can be proved for a 5-tuple set, I would like to expand the proof to be any N-tuple set of palindromes.

Likewise, all that remains to be defined mathematically is what is a palindrome, right? I am using the definition found on Wikipedia:

https://en.wikipedia.org/wiki/Palindromic_number#Formal_definition


 

Palindrome Additive Mapping Conjecture

Interestingly, if we look for the same thing but use addition as the operation that maps these sets, there are a significant subset of palindromes that seem to occur and may be a regular subset of the superset, which would imply that all the theorems applicable to group theory hold for these bijective, isomorphic mappings.

To restate the conjecture for addition we have:

 

Additive Palindrome Conjecture

,

This formulation does yield palindromic sets. Below is an excerpt of some of them.

A

B

C

B

A

Tp

1

2

3

2

1

9

8

9

10

9

8

44

19

20

21

20

19

99

80

81

82

81

80

404

82

83

84

83

82

414

84

85

86

85

84

424

86

87

88

87

86

434

88

89

90

89

88

444

90

91

92

91

90

454

92

93

94

93

92

464

94

95

96

95

94

474

96

97

98

97

96

484

98

99

100

99

98

494

181

182

183

182

181

909

183

184

185

184

183

919

185

186

187

186

185

929

187

188

189

188

187

939

189

190

191

190

189

949

191

192

193

192

191

959

193

194

195

194

193

969

195

196

197

196

195

979

197

198

199

198

197

989

199

200

201

200

199

999

800

801

802

801

800

4004

822

823

824

823

822

4114

844

845

846

845

844

4224

866

867

868

867

866

4334

888

889

890

889

888

4444

910

911

912

911

910

4554

932

933

934

933

932

4664

954

955

956

955

954

4774

976

977

978

977

976

4884

998

999

1000

999

998

4994

1801

1802

1803

1802

1801

9009

 

What this table shows is that cardinal palindromes under addition do form ordinal palindromes. They seem to occur in intervals. Further analysis will be needed to determine if this algebraic mapping does in fact result in a continuous set relation. I have begun to see a recurrent pattern already. Is there any reason why adding together numbers that are palindromic by their values should become palindromic by their order? I don't see any intrinsic reason this should be. It seems strangely related to Fibonacci's Theorem.