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