If we ask which natural numbers n are not expressible as n = ab + bc + cd (0 < a < b < c) then this is a well known open problem. Numbers not expressible in such form are called Euler's "numerus idoneus" and it is conjectured that they are finite.

If we omit the condition a < b < c and assume 0 < a <= b <= c then it was proved (assuming Generalized Riemann Hypothesis) that there is only a finite number of such numbers n.

I am interested in the problem of expressing a prime number p as p = ab + ac + bc for a >= 1 and b,c >= 2.

Anybody knows if there is some known result related to expressing prime numbers in such form? This would yield (as a corollary) a very beautiful theorem related to spanning trees in graphs.