Thanks to everyone for the explanation of why my comparison of rational
numbers did not work. I'd like to make two comments:
1. I got into this mess by trying to count points above and below the line
in the standard textbook proof of quadratic reciprocity. You have two odd
primes p and q, make the grid of integers whose x-coordinates run between 1
and (p-1)/2 and whose y-coordinates run between 1 and (q-1)/2, and draw the
line through the origin with slope q/p. I drew a beautiful picture for my
students but then miscounted the points below the line by using a rational
number comparison instead of the apparently equivalent integer comparison
(gotten by cross-multiplying). The count looked more realistic after I
cross-multiplied!
2. It occurred to me that the integers that I was dividing were of the
wrong "type." I asked sage what was going on and interpreted the output
<type 'int'>
as referring to a standard sage integer. I had no clue that I should have
been looking for
<type 'sage.rings.integer.Integer'>
Ken
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