How can we prove that this equation cannot be solved?

Question

How can we prove that this equation cannot be solved? $ 25k^3+30k^2+23k+3=x^2$ where x,k are integer numbers

Answer 1

Suppose the equation has an integer solution like a. By direct division of LHS of equation by $(k-a)$we can find that the remainder R is:

$R=23ak +30a^2+25a^3+3-x^2$

$a$ is a solution if $R=0$, that is we must have:

$R=23ak +30a^2+25a^3+3-x^2=0$

Comparing with inital equation we have:

$23ak +30a^2+25a^3=23k +30k^2+25k^3$

Since $23$ is prime to $25$ and $30$ the unique conclusion is:

$k=a=1$ which gives $x= ± 9$

That is except this solution there can be no solutions.