Strange Palindromic Number Sets

6/30/2021

Robleh Wais

The Idea

An idea to be investigated formally is specified below concerning palindromic integer number strings.

If a set has an N-tuple seed, that is constructed in such a way as to create a palindromic string, then that string can be operated upon by a binary set relation like addition. If the palindromic string is operated upon by addition does that operation create another palindrome? It does, however the difference between the initial palindromic string and the output palindromic string is that the original string may be palindromic in ordinality only, not cardinality. The output string is always palindromic in both order and value. For example, take the 5-tuple palindromic string 19,20,21,20,19, with the seed= 19,20,21. Its relational map is shown below:

19

20

21

20

19

99

 

Please note, the 5 well-ordered integers before they are summed, are palindromic in ordinal reference only. The output map in the last column is a cardinal palindromic string.

These set mappings are random and do occur as an N-tuple seed of a palindromic string in well-formed order grows.

Starting at 1 10,000 for a 5-tuple set of palindromic strings mapped via the binary operation of addition and using the palindromic number string generator A B C B A, there are 80.

See below for the results:

A

B

C

B

A

SUM

1

2

3

2

1

9

19

20

21

20

19

99

84

85

86

85

84

424

90

91

92

91

90

454

96

97

98

97

96

484

183

184

185

184

183

919

189

190

191

190

189

949

195

196

197

196

195

979

822

823

824

823

822

4114

888

889

890

889

888

4444

954

955

956

955

954

4774

1845

1846

1847

1846

1845

9229

1911

1912

1913

1912

1911

9559

1977

1978

1979

1978

1977

9889

8040

8041

8042

8041

8040

40204

8100

8101

8102

8101

8100

40504

8160

8161

8162

8161

8160

40804

8202

8203

8204

8203

8202

41014

8262

8263

8264

8263

8262

41314

8322

8323

8324

8323

8322

41614

8382

8383

8384

8383

8382

41914

8424

8425

8426

8425

8424

42124

8484

8485

8486

8485

8484

42424

8544

8545

8546

8545

8544

42724

8646

8647

8648

8647

8646

43234

8706

8707

8708

8707

8706

43534

8766

8767

8768

8767

8766

43834

8808

8809

8810

8809

8808

44044

8868

8869

8870

8869

8868

44344

8928

8929

8930

8929

8928

44644

8988

8989

8990

8989

8988

44944

9030

9031

9032

9031

9030

45154

9090

9091

9092

9091

9090

45454

9150

9151

9152

9151

9150

45754

9252

9253

9254

9253

9252

46264

9312

9313

9314

9313

9312

46564

9372

9373

9374

9373

9372

46864

9414

9415

9416

9415

9414

47074

9474

9475

9476

9475

9474

47374

9534

9535

9536

9535

9534

47674

9594

9595

9596

9595

9594

47974

9636

9637

9638

9637

9636

48184

9696

9697

9698

9697

9696

48484

9756

9757

9758

9757

9756

48784

9858

9859

9860

9859

9858

49294

9918

9919

9920

9919

9918

49594

9978

9979

9980

9979

9978

49894

18021

18022

18023

18022

18021

90109

18081

18082

18083

18082

18081

90409

18141

18142

18143

18142

18141

90709

18243

18244

18245

18244

18243

91219

18303

18304

18305

18304

18303

91519

18363

18364

18365

18364

18363

91819

18405

18406

18407

18406

18405

92029

18465

18466

18467

18466

18465

92329

18525

18526

18527

18526

18525

92629

18585

18586

18587

18586

18585

92929

18627

18628

18629

18628

18627

93139

18687

18688

18689

18688

18687

93439

18747

18748

18749

18748

18747

93739

18849

18850

18851

18850

18849

94249

18909

18910

18911

18910

18909

94549

18969

18970

18971

18970

18969

94849

19011

19012

19013

19012

19011

95059

19071

19072

19073

19072

19071

95359

19131

19132

19133

19132

19131

95659

19191

19192

19193

19192

19191

95959

19233

19234

19235

19234

19233

96169

19293

19294

19295

19294

19293

96469

19353

19354

19355

19354

19353

96769

19455

19456

19457

19456

19455

97279

19515

19516

19517

19516

19515

97579

19575

19576

19577

19576

19575

97879

19617

19618

19619

19618

19617

98089

19677

19678

19679

19678

19677

98389

19737

19738

19739

19738

19737

98689

19797

19798

19799

19798

19797

98989

19839

19840

19841

19840

19839

99199

19899

19900

19901

19900

19899

99499

19959

19960

19961

19960

19959

99799

 

That's not very many. Approximately .008 of the numbers mapped. If the mappings of palindromic sets are extended via addition the rarity is remarkable. From 200,006 additions of palindromic ordinal numbers created by a 3-tuple seed (A,B, C) of palindromes of the form A B C B A, there are only 146 palindromic cardinal strings

What becomes odder is if we take the differences in the occurrences of these cardinal palindromic number strings. There are questions we can ask. Are they cyclical? Would the differentials be palindromic? It seems based on the paucity cited above, that palindromes most likely don't occur after the 100,000 3-tuple palindromic set.

A few observations should be made concerning these number sets. Though they have been generated by an algorithmic method, their output sets are not predictable with the usual deductive method of inquiry in set theory mathematics. That is to say, we can t derive results that apply to these numbers infinitely. These numbers sets are most accessible via methods of computation suitable to software programs like Excel. No formal deductions can be extracted about them from set theory proper.