A179386 -id:A179386 - cp's OEIS frontend

A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

5, 11, 181, 207, 225, 500, 524, 1586, 13537, 376601, 223063347, 911054064, 16073515093, 22143115844, 29448160810, 1661699554612, 2498973838515, 26588790747913, 27582731314539, 178638660622364

Offset: 1

Artur Jasinski, Jul 12 2010, Jul 13 2010, Aug 03 2010

"Records of minima" means values A179386(n)=A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently A181138(y) such that A181138(y') > A181138(y) for all y' > y. See the main entry A179386 for all further considerations. - M. F. Hasler, Sep 30 2013
For d values see A179386, for x values see A179387.
Theorem (Artur Jasinski):
For any positive number x >= A179387(n), the distance between the cube of x and the square of any y (with x<>n^2 and y<>n^3) can't be less than A179386(n).
Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable there can't exist any such x (or the related y).

Cf. A179107, A179108, A179109, A179387, A179388.

Cf. A181138, A229618, A077116, A106265, A165288.

max = 1000; vecd = Table[10100, {n, 1, max}]; vecx = Table[10100, {n, 1, max}]; vecy = Table[10100, {n, 1, max}]; len = 1; min = 10100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy (*Artur Jasinski*)

A179388(n) = sqrt(A179387(n)^3 - A179386(n)).

Edited by M. F. Hasler, Sep 30 2013

A179387 Values x for "records of minima" of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

Original entry on oeis.org

3, 5, 32, 35, 37, 63, 65, 136, 568, 5215, 367806, 939787, 6369039, 7885438, 9536129, 140292677, 184151166, 890838663, 912903445, 3171881612

Offset: 1

Artur Jasinski, Jul 12 2010, Jul 13 2010, Aug 03 2010

"Records of minima" means values A154333(x) such that A154333(x') > A154333(x) for all x' > x. See the main entry A179386 for all further considerations. - M. F. Hasler, Sep 30 2013
For d values see A179386; For y values see A179388.
Theorem (Artur Jasinski):
For any positive number x >= A179387(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n).
Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and complete computable can't existed such x.
From Artur Jasinski, Aug 11 2010: (Start)
An equivalent theorem is the following (Artur Jasinski):
For any positive number x >= 1+A179387(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n+1).
(End)

Cf. A179107, A179108, A179109, A179387, A179388

max = 1000; vecd = Table[10100, {n, 1, max}]; vecx = Table[10100, {n, 1, max}]; vecy = Table[10100, {n, 1, max}]; len = 1; min = 10100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx (*Artur Jasinski*)

Edited by M. F. Hasler, Sep 30 2013

A198898 Rank of elliptic curve y^2 = x^3 - A179386(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 4

Offset: 1

Artur Jasinski, Oct 31 2011

Cf. A179386, A198897.

A179107 Records of minima of positive distance d between a square cubefree integer y and a cube of positive and squarefree integer x and such d = y^2 - x^3.

Original entry on oeis.org

1, 8, 15, 17, 24, 225, 1090, 8569, 11492, 14668, 14857, 28024, 117073

Offset: 1

Artur Jasinski, Jun 29 2010

If x=n^2 and y=n^3 distance d=0.
For x values see A179108.
For y values see A179109.
For numbers x from 46 to 108 distance can't be less than 8.
For numbers x from 109 to 5233 distance can't be less than 15.
For numbers x from 5234 to 8157 distance can't be less than 17.
For numbers x from 8158 to 729113 distance can't be less than 24.
For numbers x from 729114 to 28187350 distance can't be less than 225.
Next conjectured terms are: 4401169, 87002345, 193234265, 497218657.

J. Calvo, J. Herranz, G. Saez, A new algorithm to search for small nonzero |x^3 - y^2| values, Math. Comp. 78 (2009), 2435-2444.
N. Elkies, Hall conjecture

Cf. A179108, A179109, A179386, A179387, A179388.

d = 3; max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^d)^(1/2)] + 1; k = m^2 - n^d; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 720114}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd (* Artur Jasinski, Oct 30 2011 *)

A179407 Values x for records of minima of positive distance d between a fifth power of positive integer x and a square of integer y such d = x^5 - y^2 (x != k^2 and y != k^5).

Original entry on oeis.org

8, 55, 76, 377, 430, 499, 655, 804, 1827, 5350, 10805, 15433, 22108, 44729, 44817, 96001, 747343, 748635, 952463, 7626590, 10741787, 12798893, 14957531, 15873532

Offset: 1

Artur Jasinski, Jul 13 2010

Distance d is equal to 0 when x = k^2 and y = k^5.
For d values, see A179406.
For y values, see A179408.
Conjecture (from Artur Jasinski):
For any positive number x >= A179407(n), the distance d between the fifth power of x and the square of any y (such that x != k^2 and y != k^5) can't be less than A179406(n).

J. Blass, A Note on Diophantine Equation Y^2 + k = X^5, Math. Comp. 1976, Vol. 30, No. 135, pp. 638-640.
A. Bremner, On the Equation Y^2 = X^5 + k, Experimental Mathematics 2008 Vol. 17, No. 3, pp. 371-374.

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179406, A179408.

max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^5)^(1/2)]; k = n^5 - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 96001}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx (* Artur Jasinski, Jul 13 2010 *)

a(n)^5-A179408(n)^2 = A179406(n).

A179408 Values y for records of minima of positive distance d between a fifth power of a positive integer x and a square of an integer y such d = x^5 - y^2 (x != k^2 and y != k^5).

Original entry on oeis.org

181, 22434, 50354, 2759646, 3834168, 5562261, 10980023, 18329057, 142674503, 2093555387, 12135618855, 29588700403, 72673092233, 423129175811, 425213412449, 2855547523353, 482836315990072, 484925830443335

Offset: 1

Artur Jasinski, Jul 13 2010

Distance d is equal to 0 when x = k^2 and y = k^5.
For d values, see A179406.
For x values, see A179407.
Conjecture (from Artur Jasinski):
For any positive number x >= A179407(n), the distance d between fifth power of x and the square of any y (such that x != k^2 and y != k^5) can't be less than A179406(n).

J. Blass, A Note on Diophantine Equation Y^2 + k = X^5, Math. Comp. 1976, Vol. 30, No. 135, pp. 638-640.
A. Bremner, On the Equation Y^2 = X^5 + k, Experimental Mathematics 2008 Vol. 17, No. 3, pp. 371-374.

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179406, A179407.

max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^5)^(1/2)]; k = n^5 - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 96001}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy (* Artur Jasinski, Jul 13 2010 *)

A179407(n)^5-a(n)^2 = A179406(n).

A179784 Records for minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

Original entry on oeis.org

7, 71, 95, 448, 1756, 2215, 3983, 6271, 15231, 26775, 26870, 57475, 102703, 221916, 257963, 9053750, 9297464, 9321703, 27188154, 48787589, 62396287, 83146412, 152244535, 44475611670, 74378479183, 179884971502, 929051699593

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^7.
For x values see A179785.
For y values see A179786.
Conjecture (Artur Jasinski): For any positive number x >= A179785(n), the distance d between the seventh power of x and the square of any y (such that x <> k^2 and y <> k^7) can't be less than A179784(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179785, A179786.

d = 7; max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^d)^(1/2)]; k = n^d - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd

A179785 Values x for records of minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

Original entry on oeis.org

2, 3, 6, 8, 10, 14, 18, 20, 28, 30, 39, 55, 59, 88, 239, 255, 257, 374, 387, 477, 1136, 1221, 9104, 10959, 35962, 43783, 96569, 148544, 183163, 194933, 313592, 842163, 1254392, 1468637, 1506412, 2377393, 2407523, 4636475, 5756417, 6615968

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^7.
For d values see A179784.
For y values see A179786.
Conjecture (Artur Jasinski): For any positive number x >= A179785(n), the distance d between the seventh power of x and the square of any y (such that x <> k^2 and y <> k^7) can't be less than A179784(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179786.

d = 7; max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^d)^(1/2)]; k = n^d - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx

A179786 Values y for records of the minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

Original entry on oeis.org

11, 46, 529, 1448, 3162, 10267, 24743, 35777, 116159, 147885, 370447, 1233870, 1577546, 6392774, 211053546, 264783325, 272123427, 1011697339, 1140219273, 2370360092, 49411058753, 63606986977, 71996746561757, 137783827309893

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^7.
For d values see A179784.
For x values see A179785.
Conjecture (Artur Jasinski): For any positive number x >= A179785(n), the distance d between the seventh power of x and the square of any y (such that x <> k^2 and y <> k^7) can't be less than A179784(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785.

d = 7; max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^d)^(1/2)]; k = n^d - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy

A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

Original entry on oeis.org

93844, 322001299796379844, 1114592308630995805123571151844, 3858108676488182444301031186675778188809844, 13354661111806898918013326915229994453818137920195953844

Offset: 1

Artur Jasinski, Nov 14 2011

nonn

For y values see A200217.
For x^3-y^2 values see A200218.
The values (a(n)+1)/5 are perfect squares: for sqrt((a(n)+1)/5) see A200335.
This sequence is an infinite subset of A078933. - Artur Jasinski, Nov 27 2011

			|93844^3 - (round(sqrt(93844^3)))^2| < sqrt(93844).

For references see A078933 and A179386.

L. V. Danilov, Letter to the Editor, Mat. Zametki, 1984, Volume 36, Issue 3, Pages 457-458.
L. V. Danilov, Letter to the editor, Math. Notes 36 (3) (1984) 726.
R. D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078 [math.NT], 2014.

Cf. A078933, A179107, A179108, A179109, A179387, A179388, A199496.

Cf. A200217, A200218, A200335.

aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, xx], {n, 1, 6}]; aa
(* second program follows the generator formula *)
data = Table[(7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2 n - 1)) - (1/f)^(15 (2 n - 1))) + (1/20)*(f^(30 (2 n - 1)) + (1/f)^(30 (2 n - 1)))) /. f -> GoldenRatio, {n, 1, 6}]; data // FunctionExpand // ExpandAll // Simplify (* Bob Hanlon (hanlonr(AT)cox.net) *)
(* third program uses the Lucas numbers formula *)
Table[7/10 + (-1)^(n + 1) 3/5 LucasL[15*(2 n - 1)] +
  1/20 LucasL[30*(2 n - 1)] , {n, 1, 10}] (* Artur Jasinski, Nov 18 2011 *)

u = quadunit(20)^5
for(i=1,6, v = u^(2*i-1); t = ((-1)^i * real(v) + 57) / 125; print(5^5*t^2 - 3000*t + 719) ) \\ Noam D. Elkies

from sympy import lucas
def A200216(n): return (14+12*(lucas(k:=30*n-15) if n&1 else -lucas(k:=30*n-15))+lucas(k<<1))//20 # Chai Wah Wu, Jun 19 2024

Conjecture: a(n) = 3461450947505*a(n-1) + 6440022564931157490*a(n-2) - 6440022564931157490*a(n-3) - 3461450947505*a(n-4) + a(n-5). - R. J. Mathar, Nov 15 2011
Conjecture: g.f. 4092*(1-z)/(5*(1+1860498*z+z^2)) - 7/(10*(z-1)) + 930249*(1-z)/(10*(1-3461452808002*z+z^2)). - R. J. Mathar, Nov 15 2011
3125*A200218(n)^2 + 6750*A200218(n) + 729 = 2916*a(n). - Artur Jasinski, Nov 15 2011
125*y^2 *(54 + 50*x^3 - 25*y^2)=(9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2. - Artur Jasinski, Nov 16 2011
a(n) = 7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2*n-1))-(1/f)^(15*(2*n-1))) + (1/20)*(f^(30*(2*n-1))+(1/f)^(30*(2*n-1))), where f is golden ratio constant = (1+sqrt(5)/2). - Artur Jasinski, Nov 17 2011
a(n) = 7/10 + (3/5)*L(15*(2*n - 1))*(-1)^(n+1) + (1/20)*L(30*(2*n - 1)) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

cp's OEIS Frontend

A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

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A179387 Values x for "records of minima" of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

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A198898 Rank of elliptic curve y^2 = x^3 - A179386(n).

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A179107 Records of minima of positive distance d between a square cubefree integer y and a cube of positive and squarefree integer x and such d = y^2 - x^3.

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A179407 Values x for records of minima of positive distance d between a fifth power of positive integer x and a square of integer y such d = x^5 - y^2 (x != k^2 and y != k^5).

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A179408 Values y for records of minima of positive distance d between a fifth power of a positive integer x and a square of an integer y such d = x^5 - y^2 (x != k^2 and y != k^5).

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A179784 Records for minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

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A179785 Values x for records of minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

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A179786 Values y for records of the minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

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A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

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