$546\cdot p+546\cdot q=1001\cdot r$
$p,q$ odd primes, r positive integer.
are there infinitely many solutions?
And what if r is a Catalan number?
$\begingroup$
$\endgroup$
0
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
By Dirichlet's theorem on primes in arithmetic progressions, there are infinitely many odd primes $p$ and $q$ whose sum is divisible by $11$. For such primes $p$ and $q$, you get a solution by taking $r=6(p+q)/11$. So there are infinitely many solutions, and all solutions are of this shape. For Catalan numbers the problem seems to be out of reach.
answered Mar 3, 2023 at 13:56