Solution - Simplification or other simple results

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  p⁴³⁵⁰⁸-1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check :  p⁴³⁵⁰⁸  is the square of  p²¹⁷⁵⁴ 

Factorization is :       (p²¹⁷⁵⁴ + 1)  •  (p²¹⁷⁵⁴ - 1) 

Trying to factor as a Difference of Squares :

 1.2      Factoring:  p²¹⁷⁵⁴ - 1 

Check : 1 is the square of 1
Check :  p²¹⁷⁵⁴  is the square of  p¹⁰⁸⁷⁷ 

Factorization is :       (p¹⁰⁸⁷⁷ + 1)  •  (p¹⁰⁸⁷⁷ - 1) 

Final result :

  (1 + p21754) • (p10877 + 1) • (p10877 - 1)