On the Equation x4 + mx2y2 + y4 = z2

AbstractBy 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