Solution - Simplification or other simple results

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   a⁵²⁰m - a  =   a • (a⁵¹⁹m - 1) 

Trying to factor as a Difference of Cubes:

 2.2      Factoring:  a⁵¹⁹m - 1 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  1  is the cube of   1 
Check :  a⁵¹⁹ is the cube of   a¹⁷³

Check :  m ¹ is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result :

  a • (a519m - 1)