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.

Download the source file for this study here Palindromic Tabulations

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:

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.