cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A201047 Coordinates x of Mordell elliptic curves x^3-y^2 for primary extremal points with quadratic extensions over rationals.

Original entry on oeis.org

1942, 2878, 3862, 6100, 8380, 18694, 31228, 93844, 111382, 117118, 129910, 143950, 186145, 210025, 575800, 1193740, 1248412, 1326025, 1388545, 1501504, 1697908, 1813660, 1946737, 2069353, 2151262, 2305180, 3864190, 3897622, 54054144, 61974313, 63546025
Offset: 1

Views

Author

Artur Jasinski, Nov 26 2011

Keywords

Comments

For y coordinates see A201269.
For distances d between cubes and squares see A201268.
Primary points in A200656.
For definition primary points see A200656.
For secondary terms in A200656 see A201048.
For successive quadratic extensions see A201278.
Theorem (*Artur Jasinski*):
Every particular coordinate x contained only one extremal point.
Proof (*Artur Jasinski*): Coordinate y is computable from the formula y(x) = round(sqrt(x^3)) and distance d between cube of x and square of y is computable from the formula d(x) = x^3-(round(sqrt(x^3)))^2.

Crossrefs

Cf. A201268, A201269, A200656.

Formula

a(n) = (A201268(n)+(A201269(n))^2)^(1/3).

A202057 Numbers which are not perfect squares and such that all prime divisors are congruent to 1 or 2 mod 4.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 20, 26, 29, 32, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 97, 101, 104, 106, 109, 113, 116, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 157, 160, 164, 170, 173, 178, 181, 185, 193, 194, 197, 200, 202, 205, 208, 212, 218, 221, 226, 229, 232, 233, 241, 244, 250, 257, 260, 265, 269, 272, 274
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2011

Keywords

Comments

This sequence follows conjecture from A201278 that Mordell's elliptic curve x^3-y^2 = d can contain points {x,y} with quadratic extension sqrt(k) over rationals if and only k belongs to this sequence.
Members of A072437 that are not perfect squares. - Franklin T. Adams-Watters, Dec 15 2011

Examples

			a(3)=8 because 8 isn't perfect square and only one prime divisor 2 is congruent to 2 mod 4.

Links

Crossrefs

Cf. A004613, A201278, A072437.

Programs

  • Mathematica
    aa = {}; Do[pp = FactorInteger[j]; if = False; Do[If[Mod[pp[[n]][[1]], 4] == 3 || Mod[pp[[n]][[1]], 4] == 0, if = True], {n, 1, Length[pp]}]; If[if == False, If[IntegerQ[Sqrt[j]] == False, AppendTo[aa, j]]], {j, 2, 200}]; aa
seqQ[n_] := !IntegerQ@Sqrt[n] && AllTrue[FactorInteger[n][[;; , 1]], MemberQ[{1, 2}, Mod[#, 4]] &]; Select[Range[300], seqQ] (* Amiram Eldar, Mar 21 2020 *)

A202054 Smallest x such that Mordell's elliptic curve x^3-y^2=d (for positive d) has an integral point in the quadratic extension sqrt(A202057(n)).

Original entry on oeis.org

22, 6100, 88, 129910, 2860, 1193740, 2545, 6815614
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2011

Keywords

Comments

The existence of such x for each A202057(n) is conjectural following the conjecture in A201278.
For y values see A202055.
For d values see A202056.
a(1) = A200936(1).
a(2) = A201225(1).

Crossrefs

Cf. A201278, A202054, A202055, A202056, A202057.

A201269 Coordinates y of points {x,y} of Mordell elliptic curves x^3-y^2 for primary extremal points with quadratic extensions over rationals.

Original entry on oeis.org

85580, 154396, 240004, 476425, 767125, 2555956, 5518439, 28748141, 37172564, 40080716, 46823500, 54615700, 80311375, 96251275, 436925600, 1304261335, 1394880175, 1526959675, 1636213375, 1839881024, 2212438625, 2442495725, 2716194871, 2976815179, 3155294924
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2011

Keywords

Comments

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.

Crossrefs

Cf. A200217, A201047, A201268.

Formula

a(n) = sqrt(A201047(n)^3-A201268(n)).

A202055 Smallest y such that Mordell's elliptic curve x^3-y^2=d (for positive d) has an integral point in the quadratic extension sqrt(A202057(n)).

Original entry on oeis.org

100, 476425, 800, 46823500, 152945, 1304261335, 128375, 17793340084
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2011

Keywords

Comments

The existence of such y for each A202057(n) is conjectural following the conjecture in A201278.
For x values see A202054.
For d values see A202056.

Crossrefs

Cf. A201278, A202054, A202056, A202057.

A202056 Smallest d such that Mordell's elliptic curve x^3-y^2=d (for positive d) has an integral point in the quadratic extension sqrt(A202057(n)).

Original entry on oeis.org

648, 219375, 41472, 6021000, 1482975, 69641775, 3888000, 483568488
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2011

Keywords

Comments

The existence of such d for each A202057(n) is conjectural following the conjecture in A201278.
For x values see A202054.
For y values see A202055.

Crossrefs

Cf. A201278, A202054, A202055, A202057.

Formula

a(n) = A202054(n)^3 - A202055(n)^2
Showing 1-6 of 6 results.

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