# 1997-98NEW MEXICOHIGH SCHOOLSUPERCOMPUTINGCHALLENGE Interim Report

 Team Number: 077 School Name: MAGDALENA HIGH SCHOOL Area of Science: MATHEMATICS Project Title: CONGRUENCIES FOR PAIRWISE RELATIVELY PRIME TRIPLES Project Abstract: http://mode.lanl.k12.nm.us/97.98/abstracts/077.html Interim Report: http://mode.lanl.k12.nm.us/97.98/interims/077.html Final Report: http://mode.lanl.k12.nm.us/97.98/finalreports/077/finalreport.html

Our problem is based on a number theory question that arose from geometric research that was done by our project advisor Dr. Ben Mann of the University of New Mexico.

There is a way to associate every triple of relatively prime positive integers (p,q,r) in a geometric object. For different triples of these integers (p,q,r) and (a,b,c) one gets possibly different objects. There are different notations of equivalence for these objects.

The computation to check equivalence is in terms of congruence relations between the symmetric functions of the triple. These symmetric functions are:

• 1.Sigma(1) = p + q + r
• 2.Sigma(2) = pq + pr + qr
• 3.Sigma(3) = pqr

For two sets of triples (p,q,r) and (a,b,c) three tests of equivalence are:

• 1. Sigma(2) = Sigma(2)
• 2. Sigma(1) is congruent to +- Sigma(1) (mod 6)
• 3. Sigma(3) / Sigma(2) is congruent to Sigma(3) / Sigma(2) (by the fractional part)

We are using the fact that the Sigma(2) = Sigma(2) and the fact that each Sigma(2) is an odd number to develop a systematic search.

We are coming in after school on our own time to learn the C++ language and to work on the program. Yo is writing the program while Coyote is thinking it through and figuring out the equations (there are only two participating members). Because we have little experience in writing C++ programs it is a slow process. Before actually starting the program we outlined and planned how we were going to write the program. On our program we have derived some equations to help simplify the search so that our program will not search for numbers that cannot be possible solutions. If our program were to search through a domain of all real numbers it would be very inefficient and would take a lot more time to run on the supercomputer. We also are trying to think through our program code thoroughly before writing it so we will not have to go back and rewrite it and edit it over and over. To do this we are working very closely with our team advisor who is also teaching us to program in C++. So far we have written 74 lines of code which we have not been able to run because we are still developing more of the procedures for our program. We are pleased with the progress we have made and feel that we will be able to complete the project within the time.

We have written a function that finds sets of integers that satisfy our Sigma(2) equivalence. We have the algorithms for the Sigma(1) and Sigma(3) equivalence, but still need to write the code. We need to develop an algorithm for printing and we still need a function that tests for three relatively prime integers.

The results that we hope to accomplish are: To be able to debug and run the program successfully. To display sets of triples that pass all tests of congruency. Hopefully we will find triples that have not been discovered. Finding these undiscovered triples will also aid our project advisor in some research he is doing at the University of New Mexico.