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 a
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:
A |
B |
C |
B |
A |
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
1 |
2 |
3 |
2 |
1 |
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 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 T^{n} 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.