Abstract
A computation in Bremner is incorrect. Examples 12, 13 state that the Diophantine equations x^{4} + y^{4} = D,, with D = 4481, 5617, can have no solution in any cubic number field. The presented argument reduced the problem to a system of three simultaneous quartic equations in four variables, which were asserted to have no solution in an appropriate padic field (p = 17, 41, respectively). This latter computation is not correct, as can be seen by the following. Example: Let K = Q(θ) be the cubic number field defined by θ^{3} − θ^{2} + 10θ + 24 = 0. Then (Formula presented.) Example: Let (Formula presented.) be the cubic number field defined by φ^{3} − 7φ − 256 = 0. Then (Formula presented.) This observation regarding the examples does not affect the remaining results of the paper.
Original language  English (US) 

Number of pages  1 
Journal  Proceedings of the London Mathematical Society 
Volume  116 
Issue number  4 
DOIs 

State  Published  Apr 1 2018 
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ASJC Scopus subject areas
 Mathematics(all)
Cite this
Corrigendum to : Some Quartic Curves with no Points in any Cubic Field (Proceedings of the London Mathematical Society, (1986), s352, 2, (193214), 10.1112/plms/s352.2.193). / Bremner, Andrew.
In: Proceedings of the London Mathematical Society, Vol. 116, No. 4, 01.04.2018.Research output: Contribution to journal › Comment/debate
}
TY  JOUR
T1  Corrigendum to
T2  Some Quartic Curves with no Points in any Cubic Field (Proceedings of the London Mathematical Society, (1986), s352, 2, (193214), 10.1112/plms/s352.2.193)
AU  Bremner, Andrew
PY  2018/4/1
Y1  2018/4/1
N2  A computation in Bremner is incorrect. Examples 12, 13 state that the Diophantine equations x4 + y4 = D,, with D = 4481, 5617, can have no solution in any cubic number field. The presented argument reduced the problem to a system of three simultaneous quartic equations in four variables, which were asserted to have no solution in an appropriate padic field (p = 17, 41, respectively). This latter computation is not correct, as can be seen by the following. Example: Let K = Q(θ) be the cubic number field defined by θ3 − θ2 + 10θ + 24 = 0. Then (Formula presented.) Example: Let (Formula presented.) be the cubic number field defined by φ3 − 7φ − 256 = 0. Then (Formula presented.) This observation regarding the examples does not affect the remaining results of the paper.
AB  A computation in Bremner is incorrect. Examples 12, 13 state that the Diophantine equations x4 + y4 = D,, with D = 4481, 5617, can have no solution in any cubic number field. The presented argument reduced the problem to a system of three simultaneous quartic equations in four variables, which were asserted to have no solution in an appropriate padic field (p = 17, 41, respectively). This latter computation is not correct, as can be seen by the following. Example: Let K = Q(θ) be the cubic number field defined by θ3 − θ2 + 10θ + 24 = 0. Then (Formula presented.) Example: Let (Formula presented.) be the cubic number field defined by φ3 − 7φ − 256 = 0. Then (Formula presented.) This observation regarding the examples does not affect the remaining results of the paper.
UR  http://www.scopus.com/inward/record.url?scp=85045153713&partnerID=8YFLogxK
UR  http://www.scopus.com/inward/citedby.url?scp=85045153713&partnerID=8YFLogxK
U2  10.1112/plms.12078
DO  10.1112/plms.12078
M3  Comment/debate
AN  SCOPUS:85045153713
VL  116
JO  Proceedings of the London Mathematical Society
JF  Proceedings of the London Mathematical Society
SN  00246115
IS  4
ER 