A solution to a problem and the Diophantine equation X^2+bX+c=Y^2
  We prove that for given integers b and c, the diophantine equation x^2+bx+c=y^2, has finitely many integer solutions(i.e. pairs in ZxZ),in fact an even number of such solutions(including the zero or no solutions case).We also offer an explicit description of the solution set. Such a description depends on the form of the integer b^2-4c. Some Corollaries do follow. Furthermore, we show that the said equation has exactly two integer solutions, precisely when b^2-4c= 1,4,16,-4,or-16. In each case we list the two solutions in terms of the coefficients b and c. 
Konstantine "Hermes" Zelator