1)if p is bigger than 3 and prime,than prove $\displaystyle p^2+p$is prime.
2)$\displaystyle \forall{n}\in{N},2^{2^n}+1$is prime prove.
$\displaystyle p^2+p$ is never prime, because it factorises as $\displaystyle p(p+1)$.
$\displaystyle 2^{2^5}+1 = 4294967297 = 641\times 6700417$, so that is not prime either. It is not known whether there is any value of n greater than 4 for which the Fermat number $\displaystyle 2^{2^n}+1$ is prime.
Already done: Perfect Numbers
For separate questions, please make new threads.