On Ramanujan’s curious equality for √2(1−3−2)(1−7−2)(1−11−2)⋯\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots}

In Ramanujan’s Notebooks, Vol IV, p.20, there is the rather curious relation for primes of form $4n-1$,

Berndt asks: if this is an isolated result, or are there others? After some poking with Mathematica, it turns out that, together with $p= 2$, we can use the primes of form $4n+1$,

(Now why did Ramanujan miss this $4n+1$ counterpart?) More generally, given,

Q: Let $p =a+b+c,\;q = a b + a c + b c,\;r =abc$. For the special case $m = 2$, are there infinitely many integers $1 such that,

and $n$ is an integer? (For general $m$, see T. Andrew's comment below.)

Note: A search with Mathematica reveals numerous solutions, even for prime $a,b,c$. It is highly suggestive there may be in fact parametric solutions.

so, function m decrease monotonously ,when all values are bigger than 2. If whatever m is, this equation has only finite solutions(when all are integer), $2\leq m\leq12(n=2,a=2,b=3,c=5)$.