While reading Dick Lipton's blog, I stumbled across the following fact near the end of his Bourne Factor post:If, for every $n$, there exists a relation of the form  $$ (2^n)! = \sum_{k=0}^{m-1} a_k b_k^{c_k} $$  where $m = poly(n)$, and each of the $a_k$, $b_k$ and $c_k$ are $poly(n)$ in bit length, then factoring has polynomial sized circuits.In other words, the $(2^n)!$, which has an exponential number of bits, can potentially be represented efficiently.I have a few questions:Could someone provide a proof of the above relation, tell me the name and/or provide any references?If I were to give you $n$, $m$ and each of the $a_k$, $b_k$ and $c_k$, could you provide me a polynomial time algorithm to check the validity of the relation (i.e. is it in $NP$)?
$(2^n)! = \sum_{k=0}^{m-1} a_k b_k^{c_k} $?
ds.algorithms;reference request;factoring;comp number theory