Introduction

In today's corner, I will do some "experimental" mathematics. This new and exciting field of mathematics uses the unique capabilities of modern mathematical software (see [1, 2, 3] and Experimental Mathematics [www.expmath.org] for some examples of experimental mathematics at work).

Starting 100 years ago Rogers [4], Ramanujan, and Schur in Berndt [5] investigated the following continued fraction

For the moment ignoring the factor , this fraction is a natural geometric series analog of the simplest of all possible fractions for the golden ratio

Despite its innocent look, what is now called the Rogers-Ramanujan continued fraction still allows for new discoveries. (For various identities, see mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html.) We will derive some new modular identities for , mainly using the function NullSpace with an explicit setting of the Modulus option. We will not prove any of these rigorously (a difficult task), but rather will verify the correctness of the relations to hundreds or thousand of digits using Mathematica's high-precision arithmetic. The probability that thousands of digits of two not specially constructed real numbers agree is astronomically small.