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Trott's Corner
Exact Values In conclusion let us take a quick look at another fascinating facet of the RogersRamanujan continued fractions. Ramanujan observed that, interestingly, is an algebraic number for positive rational . The most celebrated instance is . The function Recognize from the NumberTheory` package allows us to identify an algebraic number given from an approximate value.
Using Recognize, we readily find and check that .
It is again straightforward to automate the search for an algebraic number. The function findAlgebraicValue does this. We first find a polynomial of maximal degree and then check the resulting root numerically to twice as many digits.
Next, we calculate the algebraic value of [11].
Unfortunately, Mathematica cannot find a radical expression of the root although it exists. No advanced Galois techniques are currently available.
Again using the previous modular equations, we can easily calculate (without timeconsuming calls to Recognize) exact values for . Next, we calculate the first 10 values.
These are the degrees of the resulting rootobjects (this is sequence A082682 of the OnLine Encyclopedia of Integer Sequences).
On a logarithmic scale, the coefficients of the polynomials seem to approach a limit distribution. The right graphic shows the zeros of the defining polynomials in the complex plane.
This concludes our discussion on new modular identities; however, we will come back to the RogersRamanujan continued fraction at a later date.


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