Largest of 5 prime factors: $13^4 - 6^5 + 5^9$

By rewriting $13^4 -6^5 + 5^9 = (13^4 - 6^4) + 5(25^4- 6^4)$ , we have the factorization $\begin{aligned} 13^4 -6^5 + 5^9 & = (13^4 - 6^4) + 5(25^4 - 6^4) \\ & = (13 - 6)(13 + 6)(13^2 + 6^2) + 5(25 - 6)(25 + 6)(25^2 + 6^2) \\ & = (7)(19)(205) + 5(19)(31)(661) \\ & = (5 \cdot 19)(7 \cdot 41 + 31 \cdot 661) \end{aligned}$ From here, we evaluate $7 \cdot 41 + 31 \cdot 661 = 20778$ , which is divisible by $2 \cdot 3$ as it is even and has digit sum $24$ , a multiple of 3. We have now identified four prime factors of $13^4 - 6^5 + 5^9$ . Since we are given that there are five total, the last one must be $20778 / 6 = 3463$ , which is the largest and thus the answer.

$\begin{aligned} 13^4-6^5+5^9 & = 13^4-(5+1)6^4+5^9 \\ & = 13^4-6^4+5(5^8-6^4) \\ & = (13^2-6^2)(13^2+6^2) + 5(5^4-6^2)(5^4+6^2) \\ & = (169-36)(169-36) + 5(625-36)(625+36) \\ & = 133(205) + 5(589)(661) \\ & = 5\big(133(41) + (589)(661)\big) \\ & = 5\big(7\cdot 19 (41) + (19\cdot 31)(661)\big) \\ & = 5\cdot 19 \big(7(41) + 31(661)\big) \\ & = 5\cdot 19 \big(287 + 20491\big) \\ & = 5\cdot 19 \left(\color{#3D99F6}20778\right) & \small \color{#3D99F6} 20778 \text{ is even hence divisible by }2 \\ & = 2 \cdot 5\cdot 19 \left(\color{#3D99F6}10389 \right) & \small \color{#3D99F6} \text{Digit sum of 10389 is 21, hence divisible by 3} \\ & = 2 \cdot 3 \cdot 5\cdot 19 \cdot \color{#3D99F6} 3463 \end{aligned}$

Therefore the largest prime factor is $\boxed{3463}$ .