Prime or Composite? Think factors

take the term mod $5$

$7^{88}+4^{57} \equiv 2^{88}+4^{57} \equiv 4^{44}+4^{57} \equiv (-1)^{44}+(-1)^{57} \equiv 1-1 \equiv 0 \ mod 5$

so, it is divisible by $5$ , but one can check $25$ would not divide the term.

$7^{88}+4^{57} = (7^{22})^{4}+4\times (4^{14})^{4}$ By Sophie Germain we can say this is $((7^{44}+4^{28})^{2}+4^{28})((7^{44}+4^{28})^{2}-4^{28})$ Both factors are bigger than $1$ so clearly it is not prime, also both factors differ by $2\times 4^{28}$ so it cannot be a Perfect Square.

$a^4 + 4b^4 = a^4 + 4a^2.b^2 + 4b4 - 4a^2.b^2 = (a^2 + 2b^2)2 - (2ab)^2 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)$

this is Sophie Germain’s identity

here, $a=7^{22},b=4^{14}$