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https://doi.org/10.3336/gm.48.1.04

Sums of biquadrates and elliptic curves

Julián Aguirre ; Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, Aptdo. 644, 48080 Bilbao, Spain
Juan Carlos Peral ; Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, Aptdo. 644, 48080 Bilbao, Spain

Puni tekst: engleski, pdf (129 KB) str. 49-58 preuzimanja: 316* citiraj
APA 6th Edition
Aguirre, J. i Peral, J.C. (2013). Sums of biquadrates and elliptic curves. Glasnik matematički, 48 (1), 49-58. https://doi.org/10.3336/gm.48.1.04
MLA 8th Edition
Aguirre, Julián i Juan Carlos Peral. "Sums of biquadrates and elliptic curves." Glasnik matematički, vol. 48, br. 1, 2013, str. 49-58. https://doi.org/10.3336/gm.48.1.04. Citirano 08.12.2021.
Chicago 17th Edition
Aguirre, Julián i Juan Carlos Peral. "Sums of biquadrates and elliptic curves." Glasnik matematički 48, br. 1 (2013): 49-58. https://doi.org/10.3336/gm.48.1.04
Harvard
Aguirre, J., i Peral, J.C. (2013). 'Sums of biquadrates and elliptic curves', Glasnik matematički, 48(1), str. 49-58. https://doi.org/10.3336/gm.48.1.04
Vancouver
Aguirre J, Peral JC. Sums of biquadrates and elliptic curves. Glasnik matematički [Internet]. 2013 [pristupljeno 08.12.2021.];48(1):49-58. https://doi.org/10.3336/gm.48.1.04
IEEE
J. Aguirre i J.C. Peral, "Sums of biquadrates and elliptic curves", Glasnik matematički, vol.48, br. 1, str. 49-58, 2013. [Online]. https://doi.org/10.3336/gm.48.1.04

Sažetak
Given the family of elliptic curves y2= x3-(1+u4) x, uQ, or equivalently y2=x3-(m4+n4)x for m,n integers, we prove that its rank over Q(u) is 2. We also show the existence of subfamilies of rank at least 3 and 4 over Q(u). Also, assuming the Parity Conjecture, we prove the existence of infinitely many curves having rank at least 5 over Q.
Performing an exhaustive search in the range 1 ≤ n < m ≤ 251000 we have found more than 1500 curves with rank 8, over 150 with rank 9, nine of rank 10 and one of rank 11. This improves previous results of Izadi, Khoshnam and Nabardi.

Ključne riječi
Elliptic curve; rank; biquadrate

Hrčak ID: 103269

URI
https://hrcak.srce.hr/103269

Posjeta: 526 *