cp's OEIS Frontend

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) ). 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.

1006003 = A014127(2).

2 is the only prime that occurs to a power greater than 1.

Conjecture (*Artur Jasinski*): If another infinite sequences with good Hall's examples occurred, it would have to contain primes from this sequence as constant divisors of the whole sequence, because parts of Danilov's infinite sequence (A200216, A200217, A200218) contain divisors of (3^A014127(1) - 3)/(A014127(1)^2).

a(21) > 10^18. - Max Alekseyev, Feb 26 2020