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I have a family of irrational numbers, say $w_k$ , and I have the hypothese, that they all are of the form $$ w_k = {a \beta + b\over c \beta +d } \qquad \qquad a,b,c,d \in \mathbb Z $$ where $\beta$ is likely some power of $e$.
I could decode a first small subset of numbers $w_k$ successfully using $\beta = e =\small \exp(1)$ , and also another subset using $\beta = e^2 $ with the help of Wolfram-Alpha ("W/A").
Now I have another subset where very likely I need $\beta=e^{0.5}$ , and so on.
As W/A cannot help much with the latter and I'd like to increase the subset of decodings I'd like to understand how I could approach such a decoding myself.

I use Pari/GP and the lindep procedure cannot work efficiently because in the denominator I have to expect that mixed term.

An example which I have solved is
$w_1 = 3.90612672462231517556062815632 ... $ finding $w_1=- {49 e^2- 315 \over 41 e^2-315} $ using W/A,

and one which I could not yet solve is
$w_2 =4.32632954361426254079031222453 ... $ again very likely implying $\beta = e^2$.

Another unsolved is
$ w_3 = 6.79282828995516187191178543614 ...$ which very likely implies now $\beta = e^{0.5}$

Q: how could I approach such a detection routine, or at least a meaningful preprocessing to adapt that numbers to the available (Pari/GP, W/A) procedures?

Footnote: this question occurs in an attempt to generalize the table shown in MO (which uses $\beta=e$) to other families of fractions which I had not looked at before.

P.S.: I've no good idea for tagging, perhaps someone can help improve the tags

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Gottfried Helms
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  • You can use these formulas. http://math.stackexchange.com/questions/1127654/parametrization-of-solutions-of-diophantine-equation – individ Mar 02 '17 at 04:20
  • @individ: thank you for the link. However I don't see the path to go from that - it seems completely unrelated to me? Could you make an answer here? – Gottfried Helms Mar 02 '17 at 10:00
  • As I understand it You have an equation of the form... $$A=\frac{ax^2+by^2}{cz^2+dv^2}$$ You can select those coefficients that give a root. If the equation is linear ... $$A=\frac{ax+by}{cx+dy}$$ $$x=b-dA$$ $$y=cA-a$$ Then you can choose solutions when they take the needed form. For example squares. – individ Mar 02 '17 at 11:11
  • @indiv: ah, I think I see the misunderstanding. What I have is an irrational number $A$ with arbitrary precision. Then I have the hypothese, that this number can be represented by a composition with powers of $e$. Now I feed $A$ into a procedure and want to be given the (most likely) integer coefficients $a,b,c,d$ which make $ A = { a\cdot e+b\over c \cdot e + d } $ Like for some simpler problems, given the number $10.424777... $ ,know that it should be something integer multiple of $\pi$ like $A= a\cdot \pi + b $ and find that $a=3, b=1$ such that I know $A=3 \pi + 1$ (most likely) – Gottfried Helms Mar 02 '17 at 12:02

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