Diophantine equation with primes

Question

In Linear Algebra, we were asked to make a matrix $A$ such that the product of $A$ and $\begin{bmatrix} 3 \\ -2 \\ -1 \end{bmatrix}$ is $\textbf{0}$. This in turn prompted me to ask, how many solutions are there to the linear equation $3 p_{1} -2 p_{2} - p_{3}=0$ if $p_{i}$ are primes and that $p_{1} \neq p_{2} \neq p_{3} \neq p_{1}$?

Obviously, if we didn't have the last restriction, we could make infinitely many trivial solutions by letting $p_{1} = p_{2} = p_{3}$. I wrote a Mathematica script to generate arbitrarily many solutions, so I suspect that there are infinitely many.

$\textbf{Question}$- can it be proved or disproved that there are infinitely many solutions to this equation with the added restrictions?

Answer 1

Don't see how anyone could prove it, but, for prime $r \geq 7,$ there seem to be plenty of primes $p,q$ such that $$ p + 2 q = 3 r. $$ The number of such representations grows with $r.$

Your question is milder, of course. You ask whether there are infinitely many triples, still allowing for some $r$ not to work.

 11      5      7  times three is 21

 23      5     11  times three is 33
 19      7     11  times three is 33
  7     13     11  times three is 33

 29      5     13  times three is 39
 17     11     13  times three is 39
  5     17     13  times three is 39

 41      5     17  times three is 51
 37      7     17  times three is 51
 29     11     17  times three is 51
 13     19     17  times three is 51
  5     23     17  times three is 51

 47      5     19  times three is 57
 43      7     19  times three is 57
 31     13     19  times three is 57
 23     17     19  times three is 57
 11     23     19  times three is 57

 59      5     23  times three is 69
 47     11     23  times three is 69
 43     13     23  times three is 69
 31     19     23  times three is 69
 11     29     23  times three is 69
  7     31     23  times three is 69

 73      7     29  times three is 87
 61     13     29  times three is 87
 53     17     29  times three is 87
 41     23     29  times three is 87
 13     37     29  times three is 87
  5     41     29  times three is 87

 83      5     31  times three is 93
 79      7     31  times three is 93
 71     11     31  times three is 93
 67     13     31  times three is 93
 59     17     31  times three is 93
 47     23     31  times three is 93
 19     37     31  times three is 93
 11     41     31  times three is 93
  7     43     31  times three is 93

101      5     37  times three is 111
 97      7     37  times three is 111
 89     11     37  times three is 111
 73     19     37  times three is 111
 53     29     37  times three is 111
 29     41     37  times three is 111
 17     47     37  times three is 111
  5     53     37  times three is 111

113      5     41  times three is 123
109      7     41  times three is 123
101     11     41  times three is 123
 97     13     41  times three is 123
 89     17     41  times three is 123
 61     31     41  times three is 123
 37     43     41  times three is 123
 29     47     41  times three is 123
 17     53     41  times three is 123
  5     59     41  times three is 123