On the Equation x4 + mx2y2 + y4 = z2 Original Research Article

By relating the title equation to an elliptic curve E and performing calculations with the L-series of E, we are able (subject to the standard conjectures) to determine solvability in rationals of the title equation for all m in the range m ≤ 3000. A wild assertion of Euler is corrected, a table of solutions given for m ≤ 200, and statistical information tabulated concerning the distribution of Mordell-Weil ranks and conjectural orders of Shafarevich-Tate groups.