Why $32$ isn't an Idoneal number? 
Definition :
Idoneal numbers are the positive integers $D$ such that any integer expressible in only one way as $x^2 ± Dy^2$ (where $x^2$ is relatively prime to $Dy^2$) is a prime, prime power, or twice one of these.

Number $32$ satisfies this definition because :
$p^2 = x^2+32 \cdot y^2$ if and only if : $p \equiv 1 \pmod {32}$ or $p \equiv 17 \pmod {32}$ , and every prime number
p is expressible in exactly one way as : $\sqrt{ x^2+32 \cdot y^2}$ ,where $\gcd(x^2,32 \cdot y^2)=1$ .
However , there is an additional condition on Idoneal numbers :

"A positive integer $D$ is idoneal iff it cannot be written as $ab+bc+ca$ for integer $a, b$, and $c$
with $0<a<b<c$."

Since , $32 = 1 \cdot 2 + 2 \cdot 10 +1 \cdot 10$ number $32$ doesn't satisfy this condition and therefore it isn't  Idoneal number .
My question : Why definition of Indoneal numbers is inconsistent with  the $abc$ requirement  in case of number $32$ ?
 A: The discussion so far is a little backwards -- the question of non-idoneality is not about representability of primes.  To prove non-idoneality of $D$, one has to demonstrate an $n$ which is uniquely expressible in the form $x^2+Dy^2$ which is not a prime power or twice a prime power.  
In your case, $D=32$ is not idoneal because the $n=33$ is not a prime, prime power, or twice a prime power but is expressible uniquely in the form $x^2+32y^2$.  As you observe, this is consistent with this alternative characterization of idoneal numbers in terms of expressibility by the form $ab+ac+bc$.
A: Note that you have NOT proved that EVERY prime number $p$ is uniquely expressible as $\sqrt{x^2+32y^2}$, where $(x^2,32y^2)=1$. You've simply stated what you wish to prove $:)$
While the relation 

$p^2 = x^2+32 \cdot y^2$ if and only if : $p \equiv 1 \pmod {32}$ or $p \equiv 17 \pmod {32}$ 

is true, this does NOT prove the uniqueness of $x,y$ satisfying above condition for a given $p$. So you cannot claim that there is some inconsistency in the first definition of Idoneal numbers. 
