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Suppose you calculate the first few (dozen, hundred) digits of a number which you believe to be a rational number. You can calculate the continued fraction for the number and truncate after a large number:

$ 0.67272727272727745455778089309\approx[0; 1, 2, 17, 1, 69929887587, 5, 1, 1, 2, 2] $

is probably $[0; 1, 2, 17, 1]=37/55.$

I'm wondering if there is a similarly good method for finding an algebraic number, ideally one that I can use in some computer system since large numbers are hard to work with by hand.

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    Python's `mpmath` module has a `findpoly` function.2014-03-23

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Yes, PLSQ is used on the finite set $\{1, \; \lambda, \; \lambda^2, \; \ldots, \; \lambda^n \}$ in hopes of finding a polynomial with integer coefficients for which the number $\lambda$ is a root. If such is found, sometimes the apparent relation can be proved to be correct.

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    Coefficie$n$ts of Binet for$m$ulas are algebraic, right? That's one application. I guess the characteristic polynomial gives clues as to the extension field if one knows how to look, but this seems like something that should be done automatically rather than by hand.2012-01-31