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 an 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:

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

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 a number of digits) and x^r is the reverse of x =n digits. F(x)= x+ x^r . 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 T^p

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	T^p
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.