cp's OEIS Frontend

Conjecture (Jasiński): The numbers in this sequence are multiplicative combinations of: primes congruent to 1 or 2 modulo 4 (A002313), Pythagorean primes (A002144), the number 2, and norms of Gaussian primes A055025.

Definition: Extremal points on the Mordell elliptic curve x^3 - y^2 = d are points (x,y) such that x^3 - round(sqrt(x^3))^2 = d. For values d for successive x independent of the extensions see A077119.

For y values see A200657.

For d values see A200658.

Definition: Secondary terms occur when there exist integers k such that A200656 is divisible by k^2, A200657 is divisible by k^3 and A200658 is divisible by k^6.

Terms free of such k are primary terms; see A201047. Secondary terms are, e.g., a(6)=a(2)*2^2, a(7)=a(3)*2^2, a(17)=a(10)*2^2, a(18)=a(11)*2^2, a(19)=a(12)*2^2, a(21)=a(10)*3^2.

For successive secondary terms, see A201048.

A200216 is a subsequence of this sequence.

a(1) = A200656(4) = A201047(4).

a(2) = A200656(36) = A201047(26).

All points in this sequence are extremal points (definition see A200656) and from these reason is subset of A200656 and primary (definition see A200656) and from these reason is subset of A201047.

For definition secondary terms see A200656.

For primary points in A200656 see A201047.

Secondary terms can be obtained from primary terms A201047 by ordinary multiplication by squares of particular integers while y is multiplicated by cube of that same integer and d by sixth power.

All terms in this sequence have square factor.

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.

For x coordinates see A201047.

For distances d between cubes and squares see A201268.

For successive quadratic extensions see A201278.

Theorem (*Artur Jasinski*):

Every particular coordinate y contained only one extremal point.

Proof (*Artur Jasinski*): Coordinate x is computable from the formula x(y) = round(y^(2/3)) and distance d between cube of x and square of y is computable from the formula d(y) = round(y^(2/3))^3-y^2.