Is there a set of triple-primes satisfying the following equation?


$p\_1p\_2+p\_2p\_3+p\_3p\_1+p\_1+p\_2+p\_3=2^β,p\_1p\_2p\_3=2^α-1,α>β.$
I have checked the first 11 numbers that no one satisfy the above condition. It's easy to check that when all the prime numbers $p\_i$ are in the form of $4k+3,$ then we have $p\_i=2^{s\_{i}}k\_i+2^{s\_{i}}−1$



Set of triple-primes satisfying a certain equation
Yes, there is precisely one such triple of distinct primes
with $\alpha \leq 500$, namely $(3, 11, 31)$.


It would be a surprising coincidence if there are further
such triples for larger $\alpha$.


