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For decimal numbers of degree 2. (i.e. containing a single square root), one can write the decimal as a continued fraction. If the number seems to repeat in the continued fraction we can then use some algebra to show that the decimal number is likely to be a root of a degree 2 polynomial. e.g. if the decimal was 1.41421356237 we find the continued fraction [1;2,2,2,…] and from there if we assume the 2's repeat forever we find that the number must be the solution of $x^2=2$.

I wonder if there is another procedure for decimal numbers to see if a number is likely to be the root of a cubic equation. e.g. you might be given 1.25992104989 and you want to see the smallest cubic equation with rational coefficients that would give a close value to this number.

J. W. Tanner
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zooby
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  • Your procedure for the quadratic case doesn't terminate after any fixed number of steps so here's a procedure that also doesn't terminate after any fixed number of steps: compute $x^3, x^2, x, 1$ and check all possible linear combinations of them in order of size. For less silly versions of this: https://en.wikipedia.org/wiki/Integer_relation_algorithm – Qiaochu Yuan Sep 10 '20 at 19:41
  • There is some advanced stuff to try to recover the minimal polynomial for an approximation $\bar\alpha$ of an algebraic number $\alpha$ (see §3.2 Th 7) in https://arcnl.org/jchen/download/[QFCZ09].pdf , though specifically for degree $3$ there may be simpler stuff. – zwim Sep 10 '20 at 20:08
  • See also here. – Bill Dubuque Sep 10 '20 at 22:39
  • Ah yes, one could use the LLL algorithm on $a$, $a^2$ and $a^3$. – zooby Sep 11 '20 at 01:37

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