It is known that if GRH holds there does not exist additional Idoneal numbers. (see www.mast.queensu.ca/~kani/papers/idoneal.pdf this paper puts on the question of correctnes for Wikipedia and Wolfram MathWorld since they state there could only be ONE additional Idoneal number)

What I am interested in is whether there are any certain properties that an additional idoneal number X should satisfy. I am trying to prove a theorem which would only work if X has three odd divisors. Is anything like that known or easy to derive?

I would like to find a precise reference to the fact that any idoneal number not in the currently known finite list has to have at least three odd prime factors (this should hold, as long as the answer by Pete is valid. Pete also suggested looking into the book primes of the form x^2 + ny^2 but my knowledge of number theory is too limited to derive the stated fact from there)

**EDIT**: Removed misinterpreted sentence about GRH and idoneal numbers. Added request for reference. I will give 100 bounty points to the first concise reference of this fact.