



 
Chris, I am no expert in number theory. Although we can answer specific questions, we are not much help with vague speculation. It looks as if you have an interest in the type of problems that arise in elementary number theory classes, so why not get hold of a textbook for such a course and work your way through the book. My guess is that your questions can be reduced to problems that can be solved with modular arithmetic combined with some knowledge of quadratic reciprocity. This is a field that benefits from a systematic approach (plus a lot of experimentation, either the old fashioned way with pencil and paper, or the modern way with a computer). Chris
Andrew, I worked with Chris and Harley at the University of Regina a long time ago. It doesn't look to me as if adding a third term will help much. See what you make If you look at c^{2}  c.a + a^{2} == c^{2}  c.b + b^{2} (mod p), then after rearranging If you look at a^{2} + a.b + b^{2} == c^{2}  c.a + a^{2}, then after rearranging you In order for the (triple) congruence to hold, one of each pair of conditions must In the second case, the sum of the three terms becomes 9a^{2}, which is In the first case you have a+b == c (mod p), so the sum of three terms Good luck!
 


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 