TITLE: Number Theory Representation of Primes
QUESTION [5 upvotes]: For a primes $p$ sufficiently large, does there always exists 
positive integers $k,a,b\in\mathbb{N}$ such that $p=(k+1)(ab)+k(a+b)$ or equivalently
$p\equiv (ab)\bmod ((a+b)+ab)$?
Please note that from Frobeneius $p=(k+1)X+kY$ is solvable in non-negative integers $X,Y$
provided p>XY-(X+Y), what I am looking for is the existence of a sub-class of positive intger solutions where X=ab and Y=a+b.

REPLY [3 votes]: Thanks for your interesting comments particularly about the Sophie Gemaine Primes. It is interesting that when $2p+1$ is composite with $2p+1=cd$ where $c\equiv 1(\bmod 4)$ and 
$d\equiv 3\bmod 4)$ there is always a solution. Another avenue of exploration I have been thinking about is to use some argument around constructing Covering Systems of Congruences, but this is a vague idea at best.