A201268 Distances d=x^3-y^2 for primary extremal points {x,y} of Mordell elliptic curves with quadratic extensions over rationals.

52488, 15336, -20088, 219375, -293625, -474552, 1367631, -297, 100872, -105624, 6021000, -6615000, 40608000, -45360000, -423360000, 69641775, -72560097, 110160000, -114912000, -1216512, 1418946687, -1507379625, 1450230912, -1533752064, 2143550952, 4566375

Offset

1,1

Comments

For successive x coordinates see A201047.

For successive y coordinates see A201269.

One elliptic curve with particular d can contain a finite number of extremal points.

Theorem (*Artur Jasinski*):

One elliptic curve cannot contain more than 1 extremal primary point with quadratic extension over rationals.

Consequence of this theorem is that any number in this sequence can't appear more than 1 time.

Conjecture (*Artur Jasinski*):

One elliptic curve cannot contain more than 1 point with quadratic extension over rationals.

Mordell elliptic curves contained points with extensions which are roots of polynomials : 2 degree (with Galois 2T1), 4 degree (with Galois 4T3) and 6 degree (with not soluble Galois PGL(2,5) <most of points {x,y} belonging here and rest are only rare exceptions>). Order of minimal polynomial of any extension have to divided number 12. Theoretically points can exist which are roots of polynomial of 3 degree but any such point isn't known yet.

Particular elliptic curves x^3-y^2=d can contain more than one extremal point e.g. curve x^3-y^2=-297=a(8) contained 3 of such points with coordinates x={48, 1362, 93844}={A134105(7),A134105(8),A134105(9)}.

Conjecture (*Artur Jasinski*): Extremal points are k-th successive points with maximal coordinates x.

Links

Table of n, a(n) for n=1..26.

Formula

a(n) = (A201047(n))^3-(A201269(n))^2.

Crossrefs

Cf. A200218, A201047, A201269.

Sequence in context: A263123 A235237 A237138 * A200658 A303266 A355309

Adjacent sequences: A201265 A201266 A201267 * A201269 A201270 A201271

Keyword

sign

Author

Artur Jasinski, Nov 29 2011

Status

approved