TY - JOUR
T1 - On the equation x4 + m x2y2 + y4 = z2
AU - Bremner, Andrew
AU - Jones, John
PY - 1995/2
Y1 - 1995/2
N2 - 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.
AB - 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.
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U2 - 10.1006/jnth.1995.1021
DO - 10.1006/jnth.1995.1021
M3 - Article
AN - SCOPUS:24344442537
SN - 0022-314X
VL - 50
SP - 268
EP - 298
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 2
ER -