A solution to a problem and the Diophantine equation X2+bX+c=Y2
Abstract
We prove that for given integers b and c, the diophantine equation x2+bx+c=y2, 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 b2-4c. Some Corollaries do follow. Furthermore, we show that the said equation has exactly two integer solutions, precisely when b2-4c= 1,4,16,-4,or-16. In each case we list the two solutions in terms of the coefficients b and c.
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