Impossible Discriminant

Relevant wiki: Quadratic Diophantine Equations - Modular Arithmetic Considerations

If the equation is $a{x}^{2}+bx+c=0$ , the discriminant is ${b}^{2}-4ac$ . If $b$ is even, the discriminant is divisible by $4$ . If b is odd, it can be expressed as $2m+1$ , and the discriminant will be $4{m}^{2}+4m+1-4ac$ , so it has remainder 1 when divided by $4$ .

Therefore, for an integer $D$ to be the discriminant of a quadratic equation with integer coefficients, it is necessary that $D\equiv 0, 1 \pmod4$ . Thus $\boxed { 23 }$ cannot be a discriminant.