Click on the download link at the Pari website ( http://pari.math.u-bordeaux.fr/ ) at the math department of the University of Bordeaux in France, for Pari, a free symbolic calculator similar to ones used by number theory scientists like Henri Darmon. The Pari download includes an excellent user manual and a 46 page tutorial to get you started experimenting with numbers. Be sure to look at the tutorial right away. It gives examples of things to type into Pari to immediately discover what it can do. Functions exist for number fields and elliptic curves among many other number theory tools.
Here are a few fun calculations you can do with Pari that illustrate the power and mystery of elliptic curves. Without actually explaining what is going on, Darmon hopes these examples will excite some readers to learn more.
Calculate on Pari the exponential of Pi times the square root of 163 by typing the command:
from the Pari command prompt. What do you observe?
Increase the accuracy to 50 digits by typing default(realprecision, 50) and repeat the calculation. The explanation why this number, while not an integer, is extremely close to one, lies in the theory of complex multiplication...
a) Consider the elliptic curve y2 = x3 - x, and given a prime number p, let N(p) be the number of pairs of integers (x,y) with x and y between 0 and p-1, such that p divides y2 - (x3-x). Calculate the first few values of N(p), for p =3,5,7,11, 13,17,19, 23, 29. What patterns, if any, do you observe?
b) A classical theorem of Gauss asserts that an odd prime p can be written as a sum of two squares precisely when 4 divides p-1. Verify this on the first few values of p (=5, 13, 17, 29) by writing p as a2+b2 in each case, with a odd and b even.
c) What relationship do you observe between a and N(p)?
This exercise will give the reader a feeling for one of the aspects of modularity, which asserts that the N(p) can be described in terms of interesting arithmetic data.