Numbers are too large to show $65^{64}+64^{65}$ is not a prime I tried to find cycles of powers, but they are too big. Also $65^{n} \equiv 1(\text{mod}64)$, so I dont know how to use that.

$$64^{65}+65^{64} = 6^{65}+7^{64} \pmod{29}$$
$65=2 \times 28+9, 64 = 2 \times 28 +8$, and also gcd$(29,36)$ = gcd$(29,49) = 1$
Therefore by Fermat's Little Theorem 
If gcd$(a,p)= 1$, and $p$ is a prime then $a^{(p-1)} \hspace{3pt}\equiv \hspace{3pt}1 \pmod{p}$
$36^{29-1} \equiv 1 \pmod{29}, \hspace{5pt}49^{29-1} \equiv 1 \pmod{29} \hspace{3pt} \implies \hspace{3pt}
(6^{2})^{28}  \equiv 1 \pmod{29}, \hspace{5pt} (7^{2})^{28}  \equiv 1 \pmod{29}$
Therefore $6^{65} = 6^{(56+9)} \equiv 6^9 \pmod{29}, \hspace{5pt} 7^{64} = 7^{(56+8)} \equiv 7^8 \pmod{29}$
$$64^{65}+65^{64}  \equiv 6^9+7^8 \pmod{29} \hspace{5pt} \equiv 22+7 \pmod{29} \equiv 0 \pmod{29}$$
Which shows that $$29  |  (65^{64}+64^{65})$$
Therefore $65^{64}+64^{65}$ is not a prime.
