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Linear*Quadratic = Square (Posted on 2016-11-12) Difficulty: 3 of 5
Find all integers x such that (1+x)(1+x2) is a perfect square.

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Analytic Exploration. Comment 2 of 2 |

There is a solution when each bracket is k times a square. We need only consider the second bracket.

For k=2 we have for some integers p,q: (2p^2-1)^2+1=2q^2, with p=+/-2, q=+/-5, p=+/-1, q=+/-1, p=0, q=+/-1. This is essentially the same problem as the one here.

Say k is greater than 2, for example (3p^2-1)^2+1=3q^2; but then 3p^4-2p^2-q^2 = -2/3, a fraction, and so on with any larger k, thereby contradicting the choice of p and q as integers.

So there are no other solutions.

Edited on November 12, 2016, 9:12 pm
  Posted by broll on 2016-11-12 20:53:58

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