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?
We could render the number as:
$345^4+4·(2^{172})^4$
$\equiv a^4+4b^4$ which factors as
$(a^2+2ab+2b^2)·(a^2-2ab+2b^2)$.
We then have no doubt that this number is composite.