Magdalena High School
Congruencies For Pairwise Relatively Prime Triples
This is 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 positively 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 funtions are:
- Sigma(1) = p + q + r
- Sigma(2) = pq + pr + qr
- Sigma(3) = pqr
For two sets of triples (p,q,r) and (a,b,c) three tests of equivalence are:
- Sigma(2) = Sigma(2)
- Sigma(1) is congruent to +- Sigma(1) (mod 6)
- Sigma(3) / Sigma(2) is congruent to +- Sigma(3) / Sigma(2) (by the fractional part)
The way we are going to search for these triples is to loop through all triples under a fixed value for Sigma(2) = 2d + 1 = N. We are going to use the fact that the Sigma(2) = Sigma(2) and the fact that each Sigma(2) is an odd number to develop a systematic search.