Congruence and first odd primes

Question

I tried to find the solutions to this modular equation:

$3^{(5+7+11+13+17+19+\dots +p(m-3)+p(m-2))} \equiv p(m-1) \bmod p(m) $

where $p(m)$ is the m-th odd prime number(note that it's three to the power of 5+7+11+13+...).

The only solution that I have found is $m=20$ or $p(m)=73$. Do any other such $m$ exist? What is the next one?

Answer 1

I get different numbers: $p_{53} = 241$ and $p_{730692} = 11067691$ are the first two I find. I used this code (PARI/GP):

{n=4; s=0; q=5;
forprime(p=7,10^9,
  if(Mod(3,p)^(s%(p-1))==q,print(n" "p));
  n++;s+=q;q=p
)}

The standard heuristic says that there's about a 50/50 chance that the next term is less than $10^{20}.$