Team Number: 007 School Name: Albuquerque Academy Area of Science: Mathematics/Number theory Project Title: Prime Factorization of Gaussian integers Project Abstract: http://mode.lanl.k12.nm.us/97.98/abstracts/007.html Interim Report: http://mode.lanl.k12.nm.us/97.98/interims/007.html Final Report: No final report until conclusion of SCC.
Gaussian integers are complex numbers, in the form of a + bi, such that a and b are integers, and i is the imaginary number square root of -1. These numbers are also unique in the sense that when factored, they have a unique prime factorization. Actually, this is not completely true. Each number does, in fact, have four unique prime factorizations, being that the units for this method of factorization are contained within the set, [1, -1, i, -i]. However, for the purposes of this project, we will choose to ignore this insignificant detail, and focus more on finding only one of the factorizations.
Complex numbers were discovered (as much as any mathematical concept can be truly discovered) by Karl Friedrich Gauss. Do not make the mistake of crediting complex numbers to Gauss, however; mathematicians frequently used the friendly, incomprehensible numbers long before Gauss was born in order to express nth roots of numbers. Gauss, on the other hand, employed the concept of imaginary numbers in a way that no geezer had ever thought of before. He used these numbers to express complex prime factorizations, a useful technique for those interested in number theory.
Since we will be engaging in a new endeavor, we will be forced to restart our research and development. However, as many methods for factoring Gaussian integers already exist, we will study, combine, and eventually improve upon the methods of others. After coding the various efficient algorithms within our program, we will then begin the debugging process, which we anticipate will take one month to complete.
Our program will be designed to accept input in the form of a complex number which will then be analyzed. Please note, numbers such as 3 or even imaginary number -5i are acceptable because a + bi acknowleges 0 + bi or a + 0i. This input will them be run through our optimized code, which will find the Gaussian integer's prime factorization, and display it to the user, who will delight in the efficiency and speed of our program.