How to determine in polynomial time if a number is a product of two consecutive primes? How to determine in polynomial time if a number is a product of two consecutive primes?

All I can figure out is that if Cramér's conjecture is true, then we can use the AKS primality test to find $p_i &lt; \sqrt n &lt; p_{i+1}$, then check if $p_i * p_{i+1} = n$.  Is there some way to determine if a number is of this form in polynomial time that doesn't rely on any unproven assumptions?

Also, given that a number is of this form, what is the quickest way to factor it?  How fast will modern general-purpose factoring algorithms such as the quadratic sieve factor a product of consecutive primes?
If $\rm\ n = p\:q\ $ is a product of two "close" primes, i.e.  $\rm\:|p-q| &lt; n^{1/3},\:$ then $\rm\:n\:$ can be factored in polynomial time, see Robert Erra; Christophe Grenier. The Fermat factorization method revisited. 2009. See also their slides How to compute RSA keys? The Art of RSA: Past, Present, Future.
The fastest factorization method is Number Field Sieve (NFS). To factor a 1024 bit integer $N$, it takes around $2^{86}$ many steps.