On the equation x4 + m x2y2 + y4 = z2

Andrew Bremner; John Jones

doi:10.1006/jnth.1995.1021

On the equation x⁴ + m x²y² + y⁴ = z²

Andrew Bremner
, John Jones

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	268-298
Number of pages	31
Journal	Journal of Number Theory
Volume	50
Issue number	2
DOIs	https://doi.org/10.1006/jnth.1995.1021
State	Published - Feb 1995

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1006/jnth.1995.1021

Cite this

@article{a6e82cc42ca64c49bc61bb5c829317d5,

title = "On the equation x4 + m x2y2 + y4 = z2",

abstract = "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.",

author = "Andrew Bremner and John Jones",

year = "1995",

month = feb,

doi = "10.1006/jnth.1995.1021",

language = "English (US)",

volume = "50",

pages = "268--298",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "2",

}

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.

UR - https://www.scopus.com/pages/publications/24344442537

UR - https://www.scopus.com/pages/publications/24344442537#tab=citedBy

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 -

On the equation x⁴ + m x²y² + y⁴ = z²

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this