Exact Values

In conclusion let us take a quick look at another fascinating facet of the Rogers-Ramanujan 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 time-consuming calls to Recognize) exact values for . Next, we calculate the first 10 values.

These are the degrees of the resulting root-objects (this is sequence A082682 of the On-Line 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 Rogers-Ramanujan continued fraction at a later date.