# How to prove that this Proth number cannot be a prime number?

Without using a computer prove that this Proth number cannot be a prime number :

$$43373\cdot 2^{49822}+1$$

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HINT $\rm\ c = (a\!+\!1,b\!+\!1)\ |\ a\: b^{2\:k}\! + 1\$ by $\rm\:mod\ c\!\!:\ a,\!b\equiv {-}1\ \Rightarrow\ a\: b^{2\:k}\!+1 \equiv\: -\:(-1)^{2\:k}\!+1\equiv 0\:.$

Above $\rm\ \ c = (43374,3) = 3\:.\$ Hence $\rm\ a\: b^{2\:k}\! + 1\:$ prime $\rm\:\Rightarrow\ a\!+\!1,\:b\!+\!1\:$ coprime (except in some trivial degenerate cases where the gcd $\rm\:c\:$ is not a proper factor).

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Reduce it modulo $3$. Recall $2\equiv -1$ and note $43373\equiv2$ modulo $3$, plus $49822$ is even.