Disclaimer: This page is not being actively maintained, but I will try to make small changes as I find time. The translations on this site date to my graduate student days, when I was first learning about elliptic functions. Incidentally, the translations came directly from the versions published in Crelle's Journal, and not from Jacobi's Collected Works. (The Collected Works weren't readily available to me at the time.) It shouldn't be difficult to find errors in these translations -- I would certainly appreciate hearing about any that you find.Enjoy,
Eric
Jacobi communicated work in progress to others in seminars. Among
those influenced by Jacobi's seminars was Bernhard Riemann. Riemann, a
student of Gauss, was reportedly frustrated by Gauss's secretiveness about
his research and attended Jacobi's seminars to see mathematics research
in progress. (Among Riemann's collected works is an appendix to section
40 of Jacobi's Fundamenta Nova.)
[April 25, 2000] A simple program to generate four-square representations
and some sample output to test Jacobi's version of the Four Square Theorem.
(C++
program foursq.C and sample
output foursq.txt). The program finds all representations by brute
force (in this case brute force is essentially a depth first search). The
search is optimized somewhat by using congruences pare the searches for
two-square and three-square representations. (It is easy to verify that
a two-square representation does not exist for integers congruent to 3
modulo 4, and that a three-square representation does not exist for integers
congruent to 7 modulo 8.)
[April 28, 2000] A simple program to generate r-square representations. (C++ program rsq.C and sample output rsq.txt). Essentially the same algorithm is used, except that negatives and positives are handled separately. A few additional trivial cases are eliminated using congruences. The number of representations goes up rapidly as the number of squares r increases. The program will calculate all representations of 100 as a sum of 10 squares, but expect to wait a while.
[April 28, 2000] A Mathematica program to count the number of representations of (small) integers as sums of r squares. For example, the program results show that 36 has more than 100,000,000 representations as 11 squares. (See the very last entry in the sample run for details.) This number was computed exactly without producing any representations of 36 as a sum of squares. Both Mathematica program nsq.m and sample output are available for your amusement. In addition, I used the program to show that there are exactly 1,282,320,348 representations of 100 as a sus of ten squares. The computation (nine convolutions) took about two seconds. With a little thought, you should see that I could have done this using only four convolutions. (Mathematica input and a transcript of the program run).