Squaring a circle, again

Geometry Level 5

There is a unit circle with its center point at \((0,0)\) . There is a set of 4 linear functions in the form: \[\dfrac{m}{n} x , -\dfrac{m}{n} x , \dfrac{n}{m} x , -\dfrac{n}{m} x \] which intersect the circle at 8 points. There is a square that intersects the unit circle at the same 8 points with sides parallel to X and Y axis.
If \(s\) is semi-perimeter of the square then \(\lfloor { s \times 10^9 } \rfloor = \lfloor {\pi \times 10^9 } \rfloor\).

Find the set of smallest positive integer numbers \(m < n\) such that they meet the described criteria. Find all the prime factors of numbers \(m\) and \(n\). For each factor \(p_{i}\), find \(s_{i}\) which is number of primes equal to or less then \(p_{i}\). Give answer as a product of all \(s_{i}\).

Details and Assumptions:

If some prime number appears more than once as a factor use it only once in the solution, i.e. if \(m\) has prime factors: \(2^2, 3, 5\) and \(n\) has prime factors: \( 2^3, 5^2, 7\), use only numbers \(2, 3, 5, 7\) in calculating the final answer which in this case would be \(1\times 2\times 3 \times 4 = 24\). The problem is original.


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