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.
Best Answer
If X is an idoneal number, then the class group of discriminant -4X has exponent dividing 2, so the class number is equal to the number of genera (Theorem 6 in Kani's paper: for me, this is the most convenient definition), which is given by an explicit recipe in terms of the number of prime factors of X and its congruence class mod 32 (formula (3) of Kani's paper).
In particular, if X is idoneal and is a prime or twice a prime, then its class number is at most 4. But all discriminants of class number 4 have been calculated. Indeed, all discriminants of class number up to 100 have been calculated (work of M. Watkins), so a new idoneal number should have at least $6$ odd prime divisors, or something like that.
Also see Cox's book Primes of the Form x^2 + ny^2 for treatment of idoneal numbers.