Why prime factors?

Sophie Germain identity can be used to factor the number.

Identity states that -

${ a }^{ 4 }+4{ b }^{ 4 }=\left( { a }^{ 2 }+{ 2b }^{ 2 }+2ab \right) \left( { a }^{ 2 }+{ 2b }^{ 2 }-2ab \right)$

We can rewrite our equation as

$={ 625 }^{ 2 }+4\\ ={ 5 }^{ 8 }+4\\ ={ \left( { 5 }^{ 2 } \right) }^{ 4 }+4\cdot { 1 }^{ 4 }$

On factoring, the expression simplifies as $577\cdot 677$ .

625^2 + 4 = 625^2 + 2^2 = ( 625 + 2 )^2 - 2.2.625 = 627^2 - 50^2 = ( 627 - 50 )( 627 + 50 ) = 577 . 677

Number = 625^{2} + 4 = 25^{4} Let us consider, x^{4} + 4= x^{4} + 4.[x^{2}] + 4 - 4.[x^{2}] = [x^{2} + 2]^{2} -[2x]^4 = [x^{2} + 2 + 2x][x^{2} + 2 - 2x] Similarly, 5^{4} = ([5^{2}]^{2} + 2[5^{2}] + 2)([5^{2}]^{2} - 2[5^{2}] + 2) = 577.677 So the prime factors are 577 and 677.