Hi -

I'd like to try your computation with Pari/GP. Unfortunately I don't understand your maple-code completely. There are two calls of #-preceded expressions, #flip and #reduce, without parameters.

[update] ... I see. These are surely comments.(my god; it's early sunday noon, I seem to not to have my mind up to math already...) [/update]

[update 2] This seems to be the continued fraction-process. For irrational parameters one needs infinite expansion. Well; for quadratic characteristics they are at least periodic - but what about higher algebraicities?

I once had considered this problem and seem to have found, that

- for numbers of quadratic algebraicity the solution can be expressed by a 2x2 matrix and its eigenvalues. This followed from the observation of the usual 2x2-matrix approach for the iterative solution of convergents of continued fractions.

- for higher algebraicity of the number in question we are possibly able to express the problem using a matrix of adequate dimension, establish a periodicity this way and solve then for its eigenvalues.

However, I could not complete this idea and was not able to find a way to determine the appropriate matrices. May be one should evaluate this idea again and push it up to an actual solution. This idea had, for instance, the problem, that its intermediate steps were *not* the "best approximations" in the sense as this is meant for the usual method of evaluation of continued fractions.

It were then interesting, whether this solution would be connected in one way or another to the diagonalization-approaches to tetration...

[/update 2]

Gottfried

I'd like to try your computation with Pari/GP. Unfortunately I don't understand your maple-code completely. There are two calls of #-preceded expressions, #flip and #reduce, without parameters.

[update] ... I see. These are surely comments.(my god; it's early sunday noon, I seem to not to have my mind up to math already...) [/update]

[update 2] This seems to be the continued fraction-process. For irrational parameters one needs infinite expansion. Well; for quadratic characteristics they are at least periodic - but what about higher algebraicities?

I once had considered this problem and seem to have found, that

- for numbers of quadratic algebraicity the solution can be expressed by a 2x2 matrix and its eigenvalues. This followed from the observation of the usual 2x2-matrix approach for the iterative solution of convergents of continued fractions.

- for higher algebraicity of the number in question we are possibly able to express the problem using a matrix of adequate dimension, establish a periodicity this way and solve then for its eigenvalues.

However, I could not complete this idea and was not able to find a way to determine the appropriate matrices. May be one should evaluate this idea again and push it up to an actual solution. This idea had, for instance, the problem, that its intermediate steps were *not* the "best approximations" in the sense as this is meant for the usual method of evaluation of continued fractions.

It were then interesting, whether this solution would be connected in one way or another to the diagonalization-approaches to tetration...

[/update 2]

Gottfried

Gottfried Helms, Kassel