Abyssinia-iffat

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.

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.