A more efficient way to search possible fractions close to π is via continued fraction. Then you can have a sequence that has increasing larger denominator wher

A more efficient way to search possible fractions close to π is via continued fraction. Then you can have a sequence that has increasing larger denominator where all are approximately π. Each of the fraction there would be closest to π in the nearby region (nearby in the sense that if you change the size of denominator slightly, and at the same time the numerator accordingly.) So this is close to what your getting at, just not with exhaustive search and an artificial constraint (say 10000)