Are there infinitely many Pythagorean triples with these constraints?

Question

The Pythagorean triple is triple $(a,b,c)$ such that $a,b,c$ are natural numbers which satisfy the identity $a^2+b^2=c^2$.

Let us denote the set of prime numbers as $\mathbb P$.

The question is:

Are there infinitely many pairs of prime numbers $(p,q) \in \mathbb P \times \mathbb P$ such that for every pair there exist natural number $c(p,q)$ (I write $c(p,q)$ to denote the dependence of $c$ on $p$ and $q$) such that $(p,c(p,q),q)$ or $(q,c(p,q),p)$ is a Pythagorean triple?

Remark: I created this question in my mind maybe half an hour ago while I was waiting for my friend to send me a message on my mobile phone and somehow I believe that this is a known fact, but maybe I am wrong, am I?

Edit: I edited the question because Andre Nicolas clarified my thoughts as he stated in the comment that $c(p,q)$ cannot be a hypotenuse because if that is the case then there are no such triples. In the original question this part of the question "such that $(p,c(p,q),q)$ or $(q,c(p,q),p)$ is a Pythagorean triple" was "such that $(p,q,c(p,q))$ is a Pythagorean triple" (and that is the only change).

Answer 1

Right triangles with one leg and the hypotenuse of prime length were investigated by Dubner and Forbes.

The prime legs are listed at https://oeis.org/A048161 with the first 10000 examples at https://oeis.org/A048161/b048161.txt

The hypotenuses are listed at https://oeis.org/A067756 with the first 10001 at https://oeis.org/A067756/b067756.txt

It is conjectured that there are infinitely many of these. However, there is still no resolution of the question, are there infinitely many primes $n^2 + 1?$ I cannot imagine that any more is known about primes $(n^2 + 1)/ 2,$ where this time $n$ would be odd; evidently considered by Euler: these $n$ are listed at https://oeis.org/A002731 . Your condition actually asks about $(p^2 + 1)/ 2 = q,$ with both $p,q$ prime. No-one knows.

Answer 2

There are an infinite number of prime hypotenuses and a "smaller" infinite number of prime pairs, where $A\&C$ are both prime. These may be generated by a formula, where $n$ is a set number and $k$ is a count.

$$A=(2n-1)^2+2(2n-1)k\qquad B=2(2n-1)k+2k^2\qquad C=(2n-1)^2+2(2n-1)k+2k^2$$

$C$ takes the form of $4x+1$ $may$ be prime whenever it matches a prime requirement of $6y\pm1$.

Note: $A$ can be prime only when $n=1$ because, if $n>1$, then $A$ is composite. $A=(2n-1)^2+2(2n-1)k\quad =\quad (2n-1)(2n-1+2k)$.

Here is a sample of the double primes in $Set_1$ where $C\le 10^6$.

$ \text{>run} \\ \text{Enter highest C value? 1000000}\\ \text{K9 (highest k value in set 1) is 706 where C = 998285}\\ \text{double f( 1 , 1 ) = 3 4 5 }\\ \text{double f( 1 , 2 ) = 5 12 13 }\\ \text{double f( 1 , 5 ) = 11 60 61 }\\ \text{double f( 1 , 9 ) = 19 180 181 }\\ \text{double f( 1 , 14 ) = 29 420 421 }\\ \text{double f( 1 , 29 ) = 59 1740 1741 }\\ \text{double f( 1 , 30 ) = 61 1860 1861 }\\ \text{double f( 1 , 35 ) = 71 2520 2521 }\\ \text{double f( 1 , 39 ) = 79 3120 3121 }\\ \text{double f( 1 , 50 ) = 101 5100 5101 }\\ \text{double f( 1 , 65 ) = 131 8580 8581 }\\ \text{double f( 1 , 69 ) = 139 9660 9661 }\\ \text{double f( 1 , 90 ) = 181 16380 16381 }\\ \text{double f( 1 , 99 ) = 199 19800 19801 }\\ \text{double f( 1 , 135 ) = 271 36720 36721 }\\ \text{double f( 1 , 174 ) = 349 60900 60901 }\\ \text{double f( 1 , 189 ) = 379 71820 71821 }\\ \text{double f( 1 , 204 ) = 409 83640 83641 }\\ \text{double f( 1 , 224 ) = 449 100800 100801 }\\ \text{double f( 1 , 230 ) = 461 106260 106261 }\\ \text{double f( 1 , 260 ) = 521 135720 135721 }\\ \text{double f( 1 , 284 ) = 569 161880 161881 }\\ \text{double f( 1 , 285 ) = 571 163020 163021 }\\ \text{double f( 1 , 315 ) = 631 199080 199081 }\\ \text{double f( 1 , 320 ) = 641 205440 205441 }\\ \text{double f( 1 , 330 ) = 661 218460 218461 }\\ \text{double f( 1 , 369 ) = 739 273060 273061 }\\ \text{double f( 1 , 375 ) = 751 282000 282001 }\\ \text{double f( 1 , 410 ) = 821 337020 337021 }\\ \text{double f( 1 , 440 ) = 881 388080 388081 }\\ \text{double f( 1 , 464 ) = 929 431520 431521 }\\ \text{double f( 1 , 495 ) = 991 491040 491041 }\\ \text{double f( 1 , 515 ) = 1031 531480 531481 }\\ \text{double f( 1 , 519 ) = 1039 539760 539761 }\\ \text{double f( 1 , 525 ) = 1051 552300 552301 }\\ \text{double f( 1 , 534 ) = 1069 571380 571381 }\\ \text{double f( 1 , 545 ) = 1091 595140 595141 }\\ \text{double f( 1 , 564 ) = 1129 637320 637321 }\\ \text{double f( 1 , 575 ) = 1151 662400 662401 }\\ \text{double f( 1 , 585 ) = 1171 685620 685621 }\\ \text{double f( 1 , 590 ) = 1181 697380 697381 }\\ \text{double f( 1 , 680 ) = 1361 926160 926161 }\\ \text{Pair prime count = 42 Single prime count = 39074}\\ $