Prime Numbers are really Prime!

If $p=2$ , then $p^2+2543=2547=3^2\times 283$ has $6$ positive divisors, satisfied.

If $p=3$ , then $p^2+2543=2552=2^3\times11\times29$ has $16$ positive divisors, not satisfied.

If $p>3$ , we have $p^2\equiv 1\pmod{3}$ and $p^2\equiv 1 \pmod{8}$ .

Thus, $p^2+2543\equiv0\pmod{3}$ and $p^2+2543\equiv\pmod{8}$ .

So, $p^2+2543=2^{3+\alpha_1}.3^{1+\alpha_2}.p_3^{\alpha_3}\ldots p_k^{\alpha_k}$

Hence the number positive integers of $p^2+2543\,$ is

$\qquad(\alpha_1+4)(\alpha_2+2)(\alpha_3+1)\ldots(\alpha_k+1)\ge16$ , a contradiction.

So, $p=2$ is unique solution.