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.