Large Prime numbers Say we have $2^{690} + 345^4$ and we want to figure out whether this is a prime number. 
I feel that we could break down the numbers into their respective prime factors (prime factorization) and use modular arithmetic to calculate whether these numbers are prime numbers or not. 
$690 = 2\cdot 3 \cdot 5 \cdot 23$ and $345 = 3\cdot 5\cdot 23$
How should I proceed?

The factor lulu means is $17$. To show $$2^{690}+345^4\equiv 0\mod 17$$ you can reduce the exponents modulo $\phi(17)=16$ and the bases modulo $17$. This gives $2^2+5^4$. Since $5^2\equiv 8\mod 17$ and $5^4\equiv 8^2\equiv 13\equiv-4\mod 17$, we get $$2^2+5^4\equiv 4+(-4)=0\mod 17$$
