Question

Given that x is an integer, Kelvin conjectured that x^2>x+1 .

Answer 1

We calculate the expressions $x^{2}$ and $x+1$

for each of -5, 0, 12 and 7 to see whether the inequality holds:



for x=-5, $x^{2}=(-5)^2=25$

for x=-5 $x+1=-5+1=-4$

the inequality holds, 




for x=0, $x^{2}=(0)^2=0$

for x=0 $0+1=1$

1 is not larger that 0, so x=0 is an counterexample that  
$x^2$ is not larger than $x+1$ for all integers.


Answer: 0