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how do i prove that the product of four adjacent positive integers cannot be a perfect square number?

 Nov 24, 2018
 #1
avatar+99736 
+1

We have that 

 

(a - 1) a ( a + 1) (a + 2)      for    a ≥ 2  where a is an integer

 

(a - 1) ( a + 1) a (a + 2)

 

(a^2 - 1) ( a^2 + 2a)

 

If this were a perfect square....both factors would have to  be equal

 

But...the first factor will always be <  the second factor for any   a ≥ 2

 

So....this cannot be a perfect square

 

cool cool cool

 Nov 24, 2018
edited by CPhill  Nov 24, 2018
edited by CPhill  Feb 19, 2019

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