Pythagorean triples, primes and circles.

Question

I was working recently on Twin primes and circles.

Thanks to Greg Martin, a nice generalization was conjectured in the comments section

Consider the quarter-circle with center $0$ and radius $n$ or $2n$ (where $n$ is odd). If there's only one point on that quarter-circle with integer coordinates and those coordinates greatest common divisor is $1$, then $n$ is prime.

Testing the conjecture, I noticed a pattern of Pythagorean triples appeared.

For example, with radius $n=13$, it found only one point $(12,5)$ on the quarter circle (ignore point symmetry $5,12$)

Similarly, it found these triples

(radius, x-coord, y-coord)

(5       4        3)
(13      12       5)
(61      60       11)
(181     180      19)
(421     420      29)
(1741    1740     59)
(1861    1860     61)
(2521    2520     71)
(3121    3120     79)
(5101    5100     101)
(8581    8580     131)
...

Notice the pattern shows both the radii and y-coord's are primes!

Question

Is it possible to show there are infinitely many prime y-coord's,
thus proving two Pythagorean prime sides occurs infinitely often?

Answer 1

We begin with a formula that generates the subset of Pythagorean triple where $\space (C-B)=(r-x)\space$ is an odd square, i.e. \begin{align*} &A=y=(2n-1)^2+&&2(2n-1)k\\ &B=x= &&2(2n-1)k+2k^2\\ &C=r=(2n-1)^2+&&2(2n-1)k+2k^2 \end{align*}

We can see that the $\space y$-value of $\space Set_1\space$ within this subset contains all odd numbers greater than one, and therefore contains all primes except $\space2.\quad$ Not all radii are primes but there are infinite hypotenuse values, a sample shown here , and an infinite number of primes within these values. In the table below, you can see prime radius values outside of $\space Set_1\space$ but only $\space Set_1\space$ can contain concurrent prime $\space y$-values and $\space r$-values because $\space y\space$ is always composite if $\space n>1.\quad$ A=y factors as $$A=y=(2n-1)^2+2(2n-1)k=(2 n - 1) (2 k + 2 n - 1)$$

\begin{array}{c|c|c|c|c|c|c|} n & k=1 & k=2 & k=3 & k=4 & k=5 \\ \hline Set_1 & 3,4,5 & 5,12,13& 7,24,25& 9,40,41& 11,60,61 \\ \hline Set_2 & 15,8,17 & 21,20,29 &27,36,45 &33,56,65 & 39,80,89 \\ \hline Set_3 & 35,12,37 & 45,28,53 &55,48,73 &65,72,97 & 75,100,125 \\ \hline Set_{4} &63,16,65 &77,36,85 &91,60,109 &105,88,137 &119,120,169 \\ \hline Set_{5} &99,20,101 &117,44,125 &135,72,153 &153,104,185 &171,140,221 \\ \hline \end{array}