A179387 -id:A179387 - cp's OEIS frontend

A180139 a(n)=A179387(n)+1.

4, 6, 33, 36, 38, 64, 66, 137, 569, 5216, 367807, 939788, 6369040, 7885439, 9536130, 140292678, 184151167, 890838664, 912903446, 3171881613

Offset: 1

Artur Jasinski, Aug 12 2010

Theorem (*Artur Jasinski*):
For any positive number x >= A180139(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).
Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable, such x can't exist.
If x=n^2 and y=n^3 distance d=0.
For d values see A179386.
For y values see A179388.

			For numbers x from 4 to infinity distance can't be less than 4.
For numbers x from 6 to infinity distance can't be less than 7.
For numbers x from 33 to infinity distance can't be less than 26.
For numbers x from 36 to infinity distance can't be less than 28.
For numbers x from 38 to infinity distance can't be less than 49.
For numbers x from 66 to infinity distance can't be less than 60.
For numbers x from 137 to infinity distance can't be less than 63.
For numbers x from 569 to infinity distance can't be less than 174.
For numbers x from 5216 to infinity distance can't be less than 207.
For numbers x from 367807 to infinity distance can't be less than 307.

Cf. A179107, A179108, A179109, A179387, A179388

A078933 Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.

Original entry on oeis.org

2, 5234, 8158, 93844, 367806, 421351, 720114, 939787, 28187351, 110781386, 154319269, 384242766, 390620082, 3790689201, 65589428378, 952764389446, 12438517260105, 35495694227489, 53197086958290, 5853886516781223, 12813608766102806, 23415546067124892, 38115991067861271

Offset: 1

Dean Hickerson and Robert G. Wilson v, Dec 16 2002

nonn

Hall conjectured that the nonzero difference k = x^3 - y^2 cannot be less than C x^(1/2), for a constant C. His original conjecture, probably false, has been reformulated in the following way: For any exponent e < 1/2, a constant K_e > 0 exists such that |x^3 - y^2| > K_e x^e.

Danilov found an infinite family of solutions to |x^3 - y^2| < sqrt(x). For more detail see A200216. [Artur Jasinski, Nov 04 2011]

			|5234^3 - 378661^2| = 17 < sqrt(5234), so 5234 is in the sequence.

Noam D. Elkies, Rational points near curves and small nonzero |x^3 - y^2| via lattice reduction. Algorithmic Number Theory. Proceedings of ANTS-IV; W. Bosma, ed.; Springer, 2000; pp. 33-63.
Marshall Hall Jr., The Diophantine equation x^3 - y^2 = k, in Computers in Number Theory; A. O. L. Atkin and B. Birch, eds.; Academic Press, 1971; pp. 173-198.

Frank A. Stevenson, Table of n, a(n) for n = 1..54
S. Aanderaa, L. Kristiansen, and H. K. Ruud, Search for good examples of Hall's conjecture, Math. Comp. 87 (2018), 2903-2914.
Ismael Jimenez Calvo, Marshall Hall's conjecture.
Ismael Jimenez Calvo and G. Saez Moreno, Approximate Power roots in Z_m, Proceedings of ISC 2001 (Information Security); G. I. Davida and Y. Frankel, eds.; Springer, 2001; pp. 310-323.
I. Jiminez Calvo, J. Herranz, and G. Saez, A new algorithm to search for small nonzero |x^3-y^2| values, Math. Comp. 76 (268) (2009) 2435-2444.
L. V. Danilov, Diophantine equation x^3-y^2-k and Hall's conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.
L. V. Danilov, Letter to the editors, Mat. Zametki, 36:3 (1984), 457-458.
L. V. Danilov, Letter to the editor, Mathem. 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.
Noam D. Elkies, List of integers x,y with x<10^18, 0 < |x^3-y^2| < x^(1/2).
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.

Cf. A179108, A179387, A200216.

For[x=1, True, x++, If[Abs[x^3-Round[Sqrt[x^3]]^2] < Sqrt[x] && !IntegerQ[Sqrt[x]], Print[x]]]

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 *)

A179386 Records of minima of A154333, difference of a cube minus the next smaller square.

Original entry on oeis.org

2, 4, 7, 26, 28, 47, 49, 60, 63, 174, 207, 307, 7670, 15336, 18589, 22189, 37071, 44678, 63604, 64432

Offset: 1

Artur Jasinski, Jul 13 2010, Aug 03 2010

"Records of minima" here means values A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently, the range of m(x) = min{ A154333(x') ; x' > x }. - M. F. Hasler, Sep 27 2013
For the associated x values see A179387 (and example).
For the associated values y=max{ y | y^2 < x^3 }, see A179388.
From Artur Jasinski, Jul 13 2010: (Start)
Theorem (*Artur Jasinski*)
For any positive number x >= A179387(n) the 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: The number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable, therefore such x can't exist.
(End)
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). - Artur Jasinski, Aug 11 2010
Also: The range of b(n) = min { A181138(m) | m>n }. - M. F. Hasler, Sep 26 2013
Indeed, if k=A154333(x) is a member if this sequence A179386, then also k=A181138(y) for the corresponding y, and since there is no larger x' such that x'^3-y'^3 <= k, there cannot be a larger y' such that k=A181138(y') (since this y' would require a corresponding x' > x). Conversely, the same reasoning holds for "records of minima" in A181138. - M. F. Hasler, Sep 26 and Sep 28 2013

			For numbers x > 32, A154333(x) > 7.
For numbers x > 35, A154333(x) > 26.
For numbers x > 37, A154333(x) > 28.
For numbers x > 63, A154333(x) > 47.
For numbers x > 65, A154333(x) > 49.
For numbers x > 136, A154333(x) > 60.
For numbers x > 568, A154333(x) > 63.
For numbers x > 5215, A154333(x) > 174.
For numbers x > 367806, A154333(x) > 207.
For numbers x > 939787, A154333(x) > 307.

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.
Noam Elkies, Rational points near curves and small nonzero |x^3 - y^2| via lattice reduction arXiv:math/0005139 [math.NT], 2000.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
J. Gebel, A. Petho, G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367. MR1602064

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

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; min = 10^100; 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 = 10^100; 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}]; dd

Edited by M. F. Hasler, Sep 27 2013

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A180139 a(n)=A179387(n)+1.

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A078933 Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.

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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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A179386 Records of minima of A154333, difference of a cube minus the next smaller square.

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

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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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