Can a number $n=p_1\times p_2\times \dots\times p_k+1$ where $p_1, p_2,..., p_k$ are primes, be easily factorized? (the $k$ primes are known)
If such a number could be easily factorised, then factorising $n$ would be as easy as factorising $n-1$. Which would be as easy as factorising $n-2$. And so on. So all you would have to do would be find a number $n-k$ that you can factorise, for some reasonably small $k$, and you could leapfrog your way to a factorisation of $n$.
While not conclusive, this argument suggests, to me at least, that such a number can't be easily factorised.
Not exactly. Until we know one of the primes is 2 or not, we can't tell if this value is even or odd. Until we know if 3 is one of the primes, or an odd number of the primes are 2 mod 3, we can't tell if 3 is a factor. Unless an odd number of primes are 3 mod 4, your value won't be divisible by 4. etc. But that's in effect trial division. Also it can be prime. $2\cdot 3+1=7$ as an example.