I am using EXCEL to investigate these questions.  I will work with squares up to 1000 and with all moduli under 50.  I will then see what happens with higher powers.  If you know anything about calculating on a spread sheet you can easily set up a spread sheet to obtain the relevant data.                                                                                          

You get many of these pairs if the modulus is the sum of 2 squares (ie., 4 + 9 = 13;  4 + 25 = 29).   However, a pair of squares adding to the modulus is much less if it ithe modulus  is not the sum of 2 squares (ie., 12). 

 To discover more about moduli come to the Holiday Seminars in April
we will be investigating them.  

You may also think of some other interesting questions to investigate - if so please tell us about them.