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Suppose x = 5 2 5
5 1 2 5 − 1
= x 5 − 1
= ( x − 1 ) ( x 4 + x 3 + x 2 + x + 1 )
= ( x 4 + 9 x 2 + 1 + 6 x 3 + 6 x + 2 x 2 − 5 x 3 − 1 0 x 2 − 5 x ) ( x − 1 )
= { ( x 2 + 3 x + 1 ) 2 − 5 x ( x + 1 ) 2 } ( x − 1 )
= ( x 2 + 3 x + 1 ) 2 − { 5 1 3 ( x + 1 ) } 2 ( x − 1 )
= { x 2 + 3 x + 1 + 5 1 3 ( x + 1 ) } { x 2 + 3 x + 1 − 5 1 3 ( x + 1 ) } ( x − 1 )
Hence 5 2 5 − 1 5 1 2 5 − 1 is a composite number