03/24/2023, 11:36 PM

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

( 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