<s>[INST] Find all primes of the form Find all primes of the form $3\cdot2^{2^t-1}+1$.
For $t=1$ it works.And I was thinking that it should have some logic because the other numbers go really large and is meaningless to take them one by one.
 [/INST] For $t>1$, the expression can never be prime.
Modulo 5 we get
$3 \cdot 2^{2^t - 1} + 1 \equiv (-2) \cdot 2^{2^t - 1} + 1 \mod 5$
$\equiv - 2^{2^t} + 1 \mod 5$
Since $t>1$, write $2^t = 4\cdot 2^{t-2}$. So $2^{2^t} = (2^4)^{2^{t-2}}$ and
$- 2^{2^t} + 1 \equiv -(2^4)^{2^{t-2}} + 1 \mod 5$
$\equiv -(1^{2^{t-2}}) + 1 \equiv 0 \mod 5$
Since the expression is obviously greater than 5, it follows that it is never prime.
</s>