TITLE: For what value of m that the equation $y^2 = x^3 + m$ has no integral solutions?
QUESTION [5 upvotes]: For what value of m does equation $y^2 = x^3 + m$ has no integral solutions?

REPLY [9 votes]: Here is the solution in Ireland and Rosen (page 270). 
Suppose the equation has a solution. Then $x$ is odd. For otherwise reduction modulo 4 would imply 3 is a square modulo 4.  Write the equation as 
$$y^2+1=(x+2)(x^2-2x+4)=(x+2) ((x-1)^2+3) \ . \ \ \  (*)$$
Now since $(x-1)^2 +3$  is of the form $4n+3$ there is a prime $p$ of the form $4n+3$ dividing it and reduction of $(*)$  modulo $p$  implies that $-1$ is a square modulo $p$  which is a contradiction.