Volume 9, Issue 2
Tricks of the Trade
In and Out
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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 .
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.
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