From: "Bill Daly"
Newsgroups: sci.math.research,sci.math
Subject: Reformulation of Collatz Problem
Date: Wed, 4 Mar 1998 12:10:04 -0500
I have found a curious reformulation of the Collatz Problem, which may be of
interest to someone.
Suppose that we define a function f(x) = sum(j=1..oo, f[j]*x^j) where f[j] =
1 if j ultimately descends to 1 and f[j] = 0 otherwise. The Collatz
Conjecture is then: f(x) = x/(1-x). The Collatz rules imply:
[1] f[2n] = f[n]
[2] f[2n+1] = f[6n+4]
These can be restated as follows. We have f(x) + f(-x) = 2*sum(j=1..oo,
f[2j]*x^2j) = 2*sum(j=1..oo, f[j]*x^2j) = 2*f(x^2), thus:
[1'] f(x) + f(-x) = 2*f(x^2)
Let w = cos(pi/3) + i*sin(pi/3) be a primitive 6th root of 1. Then
sum(k=0..5, f(x*w^k)*w^2k) = 6*sum(j=1..oo, f[6j+4]*x^(6j+4)) =
6*sum(j=1..oo, f[2j+1]*x^(6j+4)) = 6x*sum(j=1..oo, f[2j+1]*(x^3)^(2j+1)) =
3x*(f(x^3)-f(-x^3)), thus
[2'] sum(k=0..5, f(x*w^k)*w^2k) = 3x*(f(x^3)-f(-x^3))
Conversely, [1'] implies [1] and [2'] implies [2].
Thus, the Collatz Conjecture is equivalent to the conjecture that the only
functions f(x) which satisfy [1'] and [2'] are f(x) = ax/(1-x) for arbitrary
a.
This does not seem any simpler to me, but perhaps someone will have an
insight.
Regards,
Bill
==============================================================================
From: rusin@math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research,sci.math
Subject: Re: Reformulation of Collatz Problem
Date: 5 Mar 1998 05:13:09 GMT
In article <6dk20t$57k$1@broadway.interport.net>,
Bill Daly wrote:
>Suppose that we define a function f(x) = sum(j=1..oo, f[j]*x^j) where f[j] =
>1 if j ultimately descends to 1 and f[j] = 0 otherwise. The Collatz
>Conjecture is then: f(x) = x/(1-x). The Collatz rules imply:
>
> [1] f[2n] = f[n]
> [2] f[2n+1] = f[6n+4]
>
>These can be restated as follows. We have f(x) + f(-x) = 2*sum(j=1..oo,
>f[2j]*x^2j) = 2*sum(j=1..oo, f[j]*x^2j) = 2*f(x^2), thus:
>
> [1'] f(x) + f(-x) = 2*f(x^2)
>
>Let w = cos(pi/3) + i*sin(pi/3) be a primitive 6th root of 1. Then
>sum(k=0..5, f(x*w^k)*w^2k) = 6*sum(j=1..oo, f[6j+4]*x^(6j+4)) =
>6*sum(j=1..oo, f[2j+1]*x^(6j+4)) = 6x*sum(j=1..oo, f[2j+1]*(x^3)^(2j+1)) =
>3x*(f(x^3)-f(-x^3)), thus
>
> [2'] sum(k=0..5, f(x*w^k)*w^2k) = 3x*(f(x^3)-f(-x^3))
>
>Conversely, [1'] implies [1] and [2'] implies [2].
>
>Thus, the Collatz Conjecture is equivalent to the conjecture that the only
>functions f(x) which satisfy [1'] and [2'] are f(x) = ax/(1-x) for arbitrary
>a.
No, no, you're doing it all wrong! See, what you have to do is make
this observation, then _hide_ it, and bring up the question in some
hidden form, and hope no one else catches on. That way they might
think it's a legitimate line of inquiry and spend a few minutes on
it! Allow me to append a sample which, unfortunately for you and me,
yielded no responses.
[I had thought we could at least get someone to observe in this way that
the set of Collatz-equivalence-classes of integers was finite. :-) ]
dave
==============================================================================
From: rusin@math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research
Subject: Complex functional analysis (?) query
Date: 19 Feb 1998 02:14:27 GMT
Let H be the family of holomorphic functions on the unit disk.
If g(z)=(az+b)/(cz+d) is, say, a fractional linear transformation from
PSL(2,Z), then I can define a linear map T_g : H -> H by
T_g(f) = f o g - f i.e. T_g(f)(z) = f(g(z)) - f(z).
Naturally, an element f in ker(T_g) is simply a g-invariant function f
on the disk. If I pick a few suitable g's, then, I can arrange it so
that the functions f which lie in the kernels of all the T_g are the
holomorphic functions on some Riemann surface, and so I will know that
the intersection of the kernels of these T_g is finite-dimensional.
I would like to know if there is a reasonable way to generalize this to
other sets of maps T. How might I go about showing that
ker(T1) \intersect ker(T2) is finite dimensional
if T1 and T2 are (linear) maps H -> H of more complicated forms such as
T(f) (z) = f(z^n) + a f(z) + b f(-z)
for example, for, say, n=2 or n=3? It no longer seems appropriate to view
the kernel as the set of maps on a quotient space, so I'd need another
framework for the previous paragraph which would admit generalizations of
this type.
dave