Square root convergents is a problem in which one must find how many fractions of the square root of 2 contain a numerator with more digits than denominator.

Since we are dealing with big numbers, we should use BigIntegers. Then, I looked for a pattern since we are given the first 8 iterations.

Respectively: \(\frac{3}{2}\), \(\frac{7}{5}\), \(\frac{17}{12}\), \(\frac{41}{29}\), \(\frac{99}{70}\), \(\frac{239}{169}\), \(\frac{577}{408}\), \(\frac{1393}{985}\)

I found one that applied for both the numerator and denominator:

Afterwards, it was trivial to code the solution:

The full solution can be found here.