An easy rule for Diophantine problems which are to be resolved quickly by integral numbers
Euler returns to the problem of making formulas of the form axx+bx+g into squares. He generalizes to try to find a and b solving axx+bx+g = zyy+hy+t. This seems to rely on an initial solution and a clever application of solutions to Pell’s equation.
He does some nice examples: (1) find all triangular numbers that are also squares, (2) find all square numbers that, diminshed by one, are triangular numbers, (3) find those triangular numbers that, when tripled, are again triangular numbers.
Original Source Citation
Mémoires de l'académie des sciences de St.-Petersbourg, Volume 4, pp. 3-17.
Opera Omnia Citation
Series 1, Volume 4, pp.406-417.