Wolfram Computation Meets Knowledge

Wolfram Summer School


Eric Rowland

Summer School

Class of 2003


In June 2003 Eric graduated from the University of California, Santa Cruz, with highest honors in Mathematics. He has now embarked on a program of graduate study at Rutgers University, where his focus will be number theory. In the Fall of 2002 he participated in the Budapest Semesters in Mathematics program for American undergraduates to take classes from eminent Hungarian professors. His hobbies include studying the philosophy of science and playing the horn.

Project: The Fibonacci Sequence mod n

I have begun several investigations into obtaining a closed-form expression for f(n), the nth Fibonacci term reduced modulo n. I have looked specifically at the cases for which f(n)=0 and f(n)=1, collecting known results on the subject with new data. Several new sequences have arisen as a result. My other major attack on the problem is via binomial coefficients mod n, which also pose many interesting (and as yet unsolved) questions.

Favorite Three-Color Cellular Automaton

Rule Chosen: 2711306330654

Reason: I generated several random three-color rules with the constraint that their base 3 expressions were periodic of period 9. For example, 2711306330654 in base 3 is 100121012 100121012 100121012. The purpose of this was to “equalize” the three colors in some sense, so that no one would dominate the others.

And indeed, the background of this rule consists of alternating black and gray stripes, and the structure is mostly white. Among the random rules generated, about 75% were Class 1 and most of the others produced simple nested patterns. From an initial black cell, rule 2711306330654 generates a triangle form against a striped background. What is most interesting is that left side of the form is uniform and predictable (with a slope of 1), while the right side of the form is chaotic with a slope of about 2.

Additional Information

Rowland, E. “Fibonacci Numbers and Binomial Coefficients Mod n.” Presentation at NKS 2004, Boston, MA, 2004. [abstract] [materials]