Unstable
Which of the expressions below is factorizable?
Q 1 ( p ) = [ p − ( 2 m + 1 ) n ] [ p − ( 2 m − 1 ) n ] . . . [ p − n ] [ p + n ] . . . [ p + ( 2 m − 1 ) n ] [ p + ( 2 m + 1 ) n ] − 1
Q 2 ( p ) = [ p − ( 2 m + 1 ) n ] 2 [ p − ( 2 m − 1 ) n ] 2 . . . [ p − n ] 2 [ p + n ] 2 . . . [ p + ( 2 m − 1 ) n ] 2 [ p + ( 2 m + 1 ) n ] 2 + 1
Note: 0 = m = n = p ∈ Z .
This is part of the series: " It's easy, believe me! "
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This is not actually a solution, but rather a generalization, that:
( x − a 1 ) ( x − a 2 ) . . . ( x − a n − 1 ) ( x − a n ) − 1 and ( x − a 1 ) 2 ( x − a 2 ) 2 . . . ( x − a n − 1 ) 2 ( x − a n ) 2 + 1 are both unfactorizable in Z [ x ] , n ∈ N ∗ and a 1 = a 2 = ⋯ = a n − 1 = a n ∈ Z .
Q 1 ( p ) and Q 2 ( p ) are just a small part in this qeneralization, making a problem more specific sometimes makes the problem harder, right?