Problems & Puzzles:
Puzzles
Puzzle 194.
The Palinpoints
Joseph L. Pe defined (see
his paper) an entertaining
functional equation:
f(r(x)) = r(f(x))
....................... (1)
 x is an
integer
 f(x) is a given
arithmetic function of x, and an integer itself
 r(x) means the digitalreverse of the
integer x
The
palinpoints are all these x values that solve the equation
(1) for a given specific function f(x).
Example:
Let f(x)=isqrt(x); isqrt(x) is
the integer square root of x. Thus (1) becomes:
isqrt(r(x))=r(isqrt(x)).......(1.a)
Palinpoints
= 1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 67 76 77 88 89 98 99 100 121 131 141 144 154 164 169 179 189 400 441 451 461 484 494 505 515 525 900 961 971 981 ...and so on
Example: x=154 then
isqrt(r(154))=isqrt(451)=21 & r(isqrt(154)=r(12)=21
Pe has obtained the earliest
palinpoints for several well known
arithmetical functions f(x) such as: x^{2}, φ(x), σ(x),
p(x).
Maybe Pe named palinpoints the solutions
of (1) because usually these are:

palindromic
numbers or
 pairs of
reversal numbers.
But not always happens b.
Question 1.
Can you establish when b. is not
followed?
***
If f = p(x), being p(x) the xth prime number, the
equation (1) becomes:
p(r(x)) = r(p(x))...........(1.b)
For (1.b) Pe has found only seven
palinpoints:
x = 1, 2, 3, 4, 5, 12 & 21 ( see EIS
A069469 )
Question 2.
Can you
find one more palinpoint
to (1.b)?
***
Some minutes before the final edition
of this page, Joseph sent two
new puzzling questions about his equation (1):
Question 3:
a) What is the minimum number
of palinpoints an arithmetical function can have? b) Are there such
functions with no palinpoints at all?
_____
For tackling the question 2 maybe you'll find
useful to know the very efficient
strategy
applied by the authors (?) of the well known "The
Nth Prime Page"
Solution:
Question 1 was solved by J. K.
Andersen:
I interpret the question as:
Assume x is a palinpoint. Determine when r(x) is not a palinpoint.
The assumption is f(r(x)) = r(f(x))
r(x) is by definition a palinpoint if and only if:
f(r( r(x) )) = r(f( r(x) ))
Let r2(x) = r(r(x)), insert the assumption and the requirement is:
f(r2(x)) = r( r(f(x)) ) = r2(f(x))
r2(x) is x with any ending 0's removed.
r(x) is a palinpoint if and only if:
f(x with ending 0's removed) = f(x) with ending 0's removed.
If neither x nor f(x) has ending 0's then it is trivially satisfied.
It can be satisfied in other cases, e.g.. x=10 is a palinpoint for f when: f(10)=20 and f(1)=2. Here r(10)=1 is also a
palinpoint.
Question 2 was solved also by J. K. Andersen!!!
The next palinpoint is x = 8114118 with p(x) =
143787341.
Later he added:
I did not know the problem in advance. I modified a primefinding C
program for the search and got the result in a few seconds running. I
continued searching for bigger palindromic solutions (and only that) with
p(i)<2^32 (meaning i<=203280221) but found none. I then realized a number
with an even number of digits could never be a palprime (except 11). I
could have stopped at 10^9 but it only took a few minutes anyway.
Wondering if others had noted p(8114118) = 143787341, I just did a
Google search on "143787341". I was surprised to see that you had noted
it, even in one of your own puzzles: puzzle 51
with no solutions yet. I saw you also submitted it to Prime Curios and
EIS. It is also at Mathworld.
It appears that the search done by
J.K.A. was simplified using the following lemma pointed out by him in
his email: "x is always a palinpoint of f
if x and f(x) are both palindromes".
In any case I wonder if this is the
claimed "next" 8th palinpoint or just  as asked  "another
palinpoint". BTW, as a curio note,
143787341 is palprime!
J.K.A. replied this to my
worries:
Maybe I was a little unclear in my earlier post.
I made an exhaustive search to x = 10 million and found the first
new solution p(8114118) = 143787341. This should be guaranteed to be the
8th solution if my program is correct. It was after this I switched to
only looking for palindromic solutions. There may still be nonpalindromic
solutions with x above 10 million.
Question 3 was solved by Frank Rubin
and J. K. Andersen:
F. R.:An answer to Part 3 occurred to me right
away. Let f(x) be 10^(abs(x)+2)2.
So f(0)=98, f(1)=998, f(2)=9998, etc. Then there are no integers x and y
for which f(x)=r(f(y)) so there cannot be any
integer n for which f(r(n))=r(f(n)). f has no
palinpoints.
J. K. A.: I don't know
whether any limitation on "arithmetical function" is intended. It is easy to construct functions with no palinpoints at all. f(x) = c for a nonpalindromic constant c has no palinpoints since: f(r(x)) = c, and r(f(x)) = r(c). I think
f(x) = 10 is the simplest function without palinpoints.
Asked by me if he could produce a
nonconstant f(x) function with the asked property, he replied a solution
similar to that one sent by Rubin:
It is still easy to construct "artificial" functions
without palinpoints e.g.. f(x) = 10^(x+1)+2.
r(f(x)) = f(r(x)) will never be satisfied if the set
of all function values has no reversible pairs, i.e.. if there is not even
a solution to r(f(x)) =f(y) for any combination of x and y.
I have not looked for "normal" functions. I don't
know what that would mean anyway. Maybe a small challenge would be a
function which can take any value.
***
