How to prove 2

Marissa Singh

Marissa Singh

Answered question

2022-05-11

How to prove 2 x 2 + x + 1 is injective on the domain and codomain of all integers

Answer & Explanation

Raelynn Parker

Raelynn Parker

Beginner2022-05-12Added 12 answers

If
(1) 2 x 2 + x + 1 = 2 y 2 + y + 1 ,
then
(2) 2 ( x 2 y 2 ) + x y = ( x y ) ( 2 x + 2 y + 1 ) = 0 ,
which implies x y = 0 or 2 x + 2 y + 1 = 0. Since the latter equation has no integer solution, we are forced to conclude that x = y

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