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+--- Thread: [UFO] "commutative number theory " f(g) = g(f) (/showthread.php?tid=1729)
[UFO] "commutative number theory " f(g) = g(f) - tommy1729 - 03/24/2023
A UFO topic ; unidentified flying observations.
( Got the tag idea from Gottfried )
Hard to put in a box what kind of math this is.
Let n > 1 be an integer.
Let f(n) and g(n) be integer function that are strictly increasing.
We want
a) f(n+1) > f(n) , g(n+1) > g(n)
b) f(g(n)) = g(f(n))
c) f and g grow faster than linear.
d) f(n) + 1 < g(n) < f(f(n)) - 1
e) f and g are not both chebychev functions nor both of the form A X^B.
f) fixpoints are not allowed : f(n) =/= n , g(n) =/= n.
Notice these conditions cannot be satisfied by both f and g being polynomials.
Hence not algebra , but rather number theory.
We have no fixpoint and the integer condition so traditional dynamics is propably not easy to apply either.
Therefore the name : commutative number theory.
The construction might not be so easy and questions about existance and uniqueness or extra conditions are just around the corner.
But lets try
f(1) = 3
f := {3,5,8,12,17,23,30,38,47,...}
f(17) = 155
This f is a polynomial of degree 2 :
3+2=5
5+3=8
8+4=12
12+5=17
etc
g :={5,9,17,29,47,73,109,155,...}
and you can check that f(g(n)) = g(f(n)) and the other conditions hold.
the sequence g might not be unique and i assume no mistake has been made.
But it is not so easy to see.
The sequence g resembles or equals this perhaps :
https://oeis.org/A034329
But it might not relate or be coincidence.
I invite you to think about this.
One of the ideas is that g(n) is just floor ( f^[r](n) ) ;
in other words g(n) is just the rounded down number of some real iterate of f(n).
But that is just a conjecture.
f(n) = 1/2 (n^2 + n + 4)
f(z) = z
has the solutions
z = 1/2 + sqrt(-15)/2
<< For those who care , it might relate ?? I know Z(sqrt(-15)) is not a UFD by heart, but this one has no integer minimum polynomial ( with integer coef ). >>
regards
tommy1729