Solution - Simplification or other simple results

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  m²⁶²⁴-1 

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 :  m²⁶²⁴  is the square of  m¹³¹² 

Factorization is :       (m¹³¹² + 1)  •  (m¹³¹² - 1) 

Trying to factor as a Difference of Squares :

 1.2      Factoring:  m¹³¹² - 1 

Check : 1 is the square of 1
Check :  m¹³¹²  is the square of  m⁶⁵⁶ 

Factorization is :       (m⁶⁵⁶ + 1)  •  (m⁶⁵⁶ - 1) 

Trying to factor as a Difference of Squares :

 1.3      Factoring:  m⁶⁵⁶ - 1 

Check : 1 is the square of 1
Check :  m⁶⁵⁶  is the square of  m³²⁸ 

Factorization is :       (m³²⁸ + 1)  •  (m³²⁸ - 1) 

Trying to factor as a Difference of Squares :

 1.4      Factoring:  m³²⁸ - 1 

Check : 1 is the square of 1
Check :  m³²⁸  is the square of  m¹⁶⁴ 

Factorization is :       (m¹⁶⁴ + 1)  •  (m¹⁶⁴ - 1) 

Trying to factor as a Difference of Squares :

 1.5      Factoring:  m¹⁶⁴ - 1 

Check : 1 is the square of 1
Check :  m¹⁶⁴  is the square of  m⁸² 

Factorization is :       (m⁸² + 1)  •  (m⁸² - 1) 

Trying to factor as a Difference of Squares :

 1.6      Factoring:  m⁸² - 1 

Check : 1 is the square of 1
Check :  m⁸²  is the square of  m⁴¹ 

Factorization is :       (m⁴¹ + 1)  •  (m⁴¹ - 1) 

Final result :

  (m1312+1)•(1+m656)•(1+m328)•(m164+1)•(m82+1)•(m41+1)•(m41-1)