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A201268 Distances d=x^3-y^2 for primary extremal points {x,y} of Mordell elliptic curves with quadratic extensions over rationals. 2
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 (list; graph; refs; listen; history; text; internal format)
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.
a(n) = (A201047(n))^3-(A201269(n))^2.
Sequence in context: A263123 A235237 A237138 * A200658 A303266 A355309
Artur Jasinski, Nov 29 2011

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