1. Is the number $4^{9}+6^{10}+3^{20}$ prime?
1. We have

$$
4^{9}+6^{10}+3^{20}=\left(2^{9}\right)^{2}+2 \cdot 2^{9} \cdot 3^{10}+\left(3^{10}\right)^{2}=\left(2^{9}+3^{10}\right)^{2}
$$

Comment. One could try to compute the remainders of this number when divided by different prime numbers, hoping to find a small number that divides our number. However, this would be in vain, since $2^{9}+3^{10}$ is a prime number.