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Shifted Pythagorean Triples

 Published on Sunday, 18th October 2020, 08:00 am; Solved by 137;
Difficulty rating: 65%

Problem 730

For a non-negative integer $k$, the triple $(p,q,r)$ of positive integers is called a $k$-shifted Pythagorean triple if $$p^2 + q^2 + k = r^2$$

$(p, q, r)$ is said to be primitive if $\gcd(p, q, r)=1$.

Let $P_k(n)$ be the number of primitive k-shifted Pythagorean triples such that $1 \le p \le q \le r$ and $p + q + r \le n$.
For example, $P_0(10^4) = 703$ and $P_{20}(10^4) = 1979$.

Define $$\displaystyle S(m,n)=\sum_{k=0}^{m}P_k(n)$$ You are given that $S(10,10^4) = 10956$.

Find $S(10^2,10^8)$