A060838 -id:A060838 - cp's OEIS frontend

A309961 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 1.

6, 7, 9, 12, 13, 15, 17, 20, 22, 26, 28, 31, 33, 34, 35, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 87, 89, 90, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 114, 115, 117, 120, 123, 130, 133, 134, 136, 139, 140, 141, 142, 143, 151

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==1, print1(k", ")))

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<6 || mwr[1]==0, return(0)); if(mwr[2]>1, return(0)); ar=ellanalyticrank(E)[1]; if(ar==0, return(0)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>1 || mwr[2]<1, return(0), mwr[1]==mwr[2] && mwr[1]==1, return(1))); error("unknown (",ar==1," on the BSD conjecture)") \\ Charles R Greathouse IV, Jan 24 2023

A060838(a(n)) = 1.

A309962 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 2.

Original entry on oeis.org

19, 30, 37, 65, 86, 91, 110, 124, 126, 127, 132, 152, 153, 163, 182, 183, 201, 203, 209, 210, 217, 218, 219, 240, 246, 254, 271, 273, 282, 296, 309, 335, 342, 345, 348, 370, 379, 390, 397, 399, 407, 420, 433, 435, 436, 446, 453, 462, 468, 469, 477, 497, 498, 506, 513, 520, 523, 554

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309963 (rank 3), A309964 (rank 4).

for(k=1, 1e3, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==2, print1(k", ")))

A060838(a(n)) = 2.

A309963 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 3.

Original entry on oeis.org

657, 854, 1020, 1122, 1241, 1267, 1330, 1339, 1426, 1482, 1554, 1798, 1853, 1892, 2015, 2310, 2346, 2574, 2763, 2771, 2805, 2869, 2914, 2947, 2977, 3036, 3383, 3445, 3465, 3526, 3894, 3913, 4002, 4209, 4290, 4362, 4706, 4711, 4830, 4921, 4930, 4977, 5025, 5053, 5074, 5193, 5256

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309964 (rank 4).

for(k=1, 5e3, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==3, print1(k", ")))

A060838(a(n)) = 3.

A309964 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 4.

Original entry on oeis.org

21691, 27937, 33193, 34706, 36667, 39331, 45353, 46299, 53265, 55298, 55335, 59295, 59690, 62628, 63147, 64001, 65683, 73963, 78604, 82290, 87653, 90489, 94681, 96139

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309963 (rank 3).

for(k=1, 5e4, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==4, print1(k", ")))

A060838(a(n)) = 4.

a(18)-a(24) from Maksym Voznyy, Jan 25 2023

A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).

Original entry on oeis.org

1, 37, 2, 2, 89, 7, 683, 18, 3, 19, 25469, 3, 3, 163, 137, 1853, 631, 3, 4, 449, 7, 11, 23417, 730511, 1872, 28747, 5, 11, 4, 4, 5353, 2538163, 15409, 53, 197, 17351, 5563, 13, 433, 2570129, 13, 1176498611, 53, 1241, 4, 25903, 15642626656646177, 14, 5, 592, 4033, 165889, 90, 181, 9109, 5266097, 5, 184223499139, 5, 5, 7, 52954777

Offset: 1

Jean-François Alcover, May 11 2011

nonn

This sequence a(k) is computed so that equation a(k)^3 + y^3 = A020898(k)*z^3 holds.
The 4 sequences A020898 [i.e., n], A190356 [i.e., x], A190580 [i.e., y] and A190581 [i.e., z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3.
All x values above 25469 were obtained from Mishima's list and may not be the least positive solution.

			a(18) = 3 because A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Cf. A020898, A060838, A190580, A190581

(* Let x = u + v and y = u - v *)
f[n_, m_] := (r =  Reduce[u > 0 && v > 0 && Mod[2*u^3 + 6*u*v^2, n] == 0, {u, v},  Integers] ;
uv={u,v}/.(ToRules/@ List@@ r[[All,-2;;-1]])/.C-> c;
xy = (s = {};
Do[sel =  Select[uv,  IntegerQ[((2*#1[[1]]^3 + 6*#1[[1]]*#1[[2]]^2)/n)^(1/ 3)] &];
If[sel =!= {}, AppendTo[s, sel] ], {c[1], 0, m}, {c[2], 0,  m}];
{#[[1]] + #[[2]], #[[1]] - #[[2]]} & /@ (s //
Flatten[#, 1] &)) // Select[#, Total[#] != 0 &] &;
nxyz =  xy /. {x_Integer, y_} -> {n, x, y, ((x^3 + y^3)/n)^(1/3)};
nxyz /. ({, x, y_, z_} /; {x, y, z} != {0, 0, 0} &&
GCD[x, y, z] != 1) :> (gd = GCD[x, y, z]; {n, x/gd, y/gd, z/gd})) // Union // Sort[#, #1[[2]] < #2[[2]] &] &;
g[n_] := (m0 = 1; While[(r = f[n, m0]) == {}, m0 = 2 m0];
r // First);
A020898 = {2, 6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 49, 50, 51, 53, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 97, 98, 103, 105, 106, 107, 110, 114, 115, 117, 123, 124, 126, 127, 130}; km = Length[A020898]; (* xm(n) = some hard to compute values of x from Hisanori Mishima's list *) xm[22]=25469; xm[50]=23417; xm[51]=730511; xm[58]=28747; xm[68]=2538163; xm[69]=15409; xm[75]=17351; xm[85]=2570129; xm[87]=1176498611; xm[92]=25903; xm[94]=15642626656646177; xm[106]=165889; xm[114]=9109; xm[115]=5266097; xm[123]=184223499139; xm[130]=52954777; xm[n_] := xm[n] = g[n][[2]];
A190356 = Table[ n = A020898[[k]]; Print[xm[n]]; xm[n], {k, 1, km}] (* Jean-François Alcover, Jan 03 2012 *)

Positions corresponding to n=124 and n=127 (which were not minimal) corrected by Jean-François Alcover

Extended to 62 terms by Jean-François Alcover, Jan 03 2012

A190580 Value of y in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

Original entry on oeis.org

1, 17, -1, 1, 19, 2, 397, -1, -2, 1, 17299, -1, 1, 107, -65, 523, -359, 2, -3, -71, 1, -2, -11267, 62641, -1819, -14653, -4, 7, -1, 1, 1208, -472663, -10441, 17, -126, -11951, 53, -4, 323, -2404889, 5, -907929611, 36, -431, 3, -3547, -15616184186396177, -5, -3, -349, 3527, -140131, 17, -71, -901, -2741617, -2, 10183412861, -1, 1, -6, 33728183

Offset: 1

Jean-François Alcover, May 13 2011

sign

A190356(n)^3 + a(n)^3 = A020898(n)*z^3. Unknown z corresponds to sequence A190581.

The 4 sequences A020898 [i.e. n], A190356 [i.e. x], A190580 [i.e. y] and A190581 [i.e. z] satisfy the equation A190356^3 + A190580^3 = A020898 * A190581^3

			a(18) = 2  because  A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Cf. A020898, A060838, A190356, A190581.

Table[ y /. First[ Solve[ A190356[[n]]^3 + y^3 == A020898[[n]] * A190581[[n]]^3 ] ], {n, 62}] (* Jean-François Alcover, Jan 04 2012 *)

A190581 Value of z in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

Original entry on oeis.org

1, 21, 1, 1, 39, 3, 294, 7, 1, 7, 9954, 1, 1, 57, 42, 582, 182, 1, 1, 129, 2, 3, 6111, 197028, 217, 7083, 1, 3, 1, 1, 1323, 620505, 3318, 13, 43, 3606, 1302, 3, 111, 330498, 3, 216266610, 13, 273, 1, 5733, 590736058375050, 3, 1, 117, 1014, 25767, 19, 37, 1878, 1029364, 1, 37045412880, 1, 1, 1, 11285694

Offset: 1

Jean-François Alcover, May 13 2011

nonn

A190356(n)^3 + y^3 = A020898(n)*a(n)^3. Unknown y corresponds to sequence A190580.

The 4 sequences A020898 [i.e. n], A190356 [i.e. x], A190580 [i.e. y] and A190581 [i.e. z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3

			a(18) = 1  because  A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Cf. A020898, A060838, A190356, A190580

a[n_] := z /. ToRules[ Reduce[ z > 0 && A190356[[n]]^3 + A190580[[n]]^3 == A020898[[n]]*z^3, z, Integers]]; Table[a[n] , {n, 1, 62}]

A228499 Sums of two rational cubes, excluding cubes and twice cubes.

Original entry on oeis.org

6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117, 120, 123, 124

Offset: 1

Arkadiusz Wesolowski, Aug 23 2013

nonn

Each term can be written as sum of two rational cubes infinitely many times.

These are all the integers A>0 such that the rank of the elliptic curve x^3 + y^3 = A is positive (A060838(A)>0). - Michael Somos, Feb 29 2020

Wacław Sierpiński, Teoria liczb, cz. II, PWN, Warsaw, 1959, pp. 472-473.

Index entries for sequences related to sums of cubes

Subsequence of A020897, and hence of A159843.

Cf. A020898, A060838.

for(n=1, 124, if(ellanalyticrank(ellinit([0, (4*n)^2]))[1]>0, print1(n, ", ")));

A230564 Rational rank of the n-th taxicab elliptic curve x^3 + y^3 = A011541(n).

Original entry on oeis.org

0, 2, 4, 5, 4

Offset: 1

Jonathan Sondow, Oct 25 2013

Guy, 2004: "Andrew Bremner has computed the rational rank of the elliptic curve x^3 + y^3 = Taxicab(n) as equal to 2, 4, 5, 4 for n = 2, 3, 4, 5, respectively."
Abhinav Kumar computed that a(1) = 0 (see the MathOverflow link for details). But Euler and Legendre scooped him (see the next comment).
Noam D. Elkies: "... the fact that x^3+y^3=2 has no [rational] solutions other than x=y=1 is attributed by Dickson to Euler himself: see Dickson's History of the Theory of Numbers (1920) Vol.II, Chapter XXI "Numbers the Sum of Two Rational Cubes", page 572. The reference (footnote 182) is "Algebra, 2, 170, Art. 247; French transl., 2, 1774, pp. 355-60; Opera Omnia, (1), I, 491". In the next page Dickson also refers to work of Legendre that includes this result (footnote 184: "Théorie des nombres, Paris, 1798, 409; ...")." See the MathOverflow link for further comments from Elkies.

			rank(x^3 + y^3 = 2) = 0.
rank(x^3 + y^3 = 1729) = 2.
rank(x^3 + y^3 = 87539319) = 4.
rank(x^3 + y^3 = 6963472309248) = 5.
rank(x^3 + y^3 = 48988659276962496) = 4.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, D1.

MathOverflow, What is the rational rank of the elliptic curve x^3 + y^3 = 2?, Oct 25 2013.
J. Silverman, Taxicabs and sums of two cubes, Amer. Math. Monthly, 100 (1993), 331-340.

Cf. A011541, A060838, A080642.

a(n) = A060838(A011541(n)).

A359687 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 5.

Original entry on oeis.org

489489, 525698, 526535, 763002, 903210, 1423214

Offset: 1

Maksym Voznyy and Charles R Greathouse IV, Jan 25 2023

Subsequence of A159843.

Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

is(n)=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<489489, return(0)); if(mwr[1]>5 || mwr[2]<5, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(0)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>5 || mwr[2]<5, return(0), mwr[1]==5 && mwr[2]==5, return(1))); Str("unknown; ",ar==5," under BSD conjecture") \\ Charles R Greathouse IV, Jan 25 2023

A060838(a(n)) = 5.

cp's OEIS Frontend

A309961 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 1.

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A309962 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 2.

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A309963 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 3.

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A309964 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 4.

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A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).

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A190580 Value of y in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

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A190581 Value of z in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

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A228499 Sums of two rational cubes, excluding cubes and twice cubes.

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A230564 Rational rank of the n-th taxicab elliptic curve x^3 + y^3 = A011541(n).