A179107 -id:A179107 - cp's OEIS frontend

A179406 Record minima of the positive distance d between the fifth power of a positive integer x and the square of an integer y such that d = x^5 - y^2 (x != k^2 and y != k^5).

7, 19, 60, 341, 47776, 70378, 78846, 115775, 220898, 780231, 2242100, 11889984, 26914479, 50406928, 77146256, 80117392, 284679759, 595974650, 2071791247, 7825152599, 67944824923, 742629277177, 1709838230002, 2676465117663

Offset: 1

Artur Jasinski, Jul 13 2010

Distance d is equal to 0 when x = k^2 and y = k^5.
For x values see A179407.
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, A179407, 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}]; dd

A179798 Record minima of the positive distance d between the 11th power of a positive integer x and the square of an integer y such that d = x^13 - y^2 (x <> k^2 and y <> k^13).

Original entry on oeis.org

92, 1679, 39281, 89927, 296863, 1530322, 12056004, 55972895, 67903894, 102383343, 641211875, 5148097536, 13764973788, 19839459725, 87957606400, 113794567580, 126889914716, 146745583311, 880304597278, 1154049177924

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^13.
For x values see A179799.
For x values see A179800.
Conjecture (Artur Jasinski):
For any positive number x >= A179799(n), the distance d between the eleventh power of x and the square of any y (such that x <> k^2 and y <> k^13) can't be less than A179798(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800.

d = 13; 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

A179799 Values x for records of minima of the positive distance d between an 11th power of a positive integer x and a square of an integer y such that d = x^13 - y^2 (x<>k^2 and y<>k^13).

Original entry on oeis.org

2, 3, 5, 6, 8, 11, 13, 14, 23, 24, 35, 40, 42, 45, 50, 54, 62, 70, 79, 85, 88, 89, 142, 152, 220, 345, 353, 364, 412, 416, 455, 627, 734, 743, 911, 921, 1068, 1095, 1294, 1894, 2398, 2719, 2887, 3015, 3623, 3814, 5837, 6226, 8603, 8669, 8971, 9987, 12683

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d = 0 when x = k^2 and y = k^13.
For d values see A179798.
For y values see A179800.
Conjecture: For any positive number x >= A179799(n) the distance d between the 11th power of x and the square of any y (such that x<>k^2 and y<>k^13) can't be less than A179798(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800.

d = 13; 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

A179800 Values y for record minima of the positive distance d between the thirteenth power of a positive integer x and the square of an integer y such that d = x^13 - y^2 (x <> k^2 and y <> k^13).

Original entry on oeis.org

90, 1262, 34938, 114283, 741455, 5875603, 17403307, 28172943, 709955183, 936209559, 10875326100, 25905378592, 35572991418, 55703353220, 110485434560, 182204642678, 447245502234, 984322154617, 2160608565081, 3477146726351

Offset: 1

Artur Jasinski, Jul 27 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^13.
For d values see A179798.
For x values see A179799.
Conjecture: For any positive number x >= A179799(n), the distance d between the 13th power of x and the square of any y (such that x <> k^2 and y <> k^13) can't be less than A179798(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800.

d = 13; 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

A179812 Record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).

Original entry on oeis.org

7, 7538, 283261, 494576, 4235622, 7135951, 38053824, 55905695, 185380312, 1208691743, 3263221507, 14034746735, 14732727599, 24211719874, 68491624661, 136264246246, 5337970328375, 6845918569200, 15505738619231, 30037885135088

Offset: 1

Artur Jasinski, Jul 28 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^15.
For x values see A179813.
For y values see A179814.
Conjecture: For any positive number x >= A179813(n), the distance d between the fifteenth power of x and the square of any y (such that x <> k^2 and y <> k^15) can't be less than A179812(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800, A179812, A179813, A179814.

d = 15; 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

A179813 Values x for record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 17, 18, 23, 24, 27, 35, 45, 55, 56, 76, 78, 84, 111, 114, 115, 117, 118, 139, 164, 172, 175, 176, 179, 183, 188, 190, 193, 305, 316, 377, 395, 461, 466, 483, 485, 654, 747, 868, 877, 931, 1045, 1434, 1822, 2199, 2645, 2754, 3171, 3961

Offset: 1

Artur Jasinski, Jul 28 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^15.
For x values see A179813.
For y values see A179814.
Conjecture: For any positive number x >= A179813(n), the distance d between the fifteenth power of x and the square of any y (such that x <> k^2 and y <> k^15) can't be less than A179812(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800, A179812, A179813, A179814.

d = 15; 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

A179814 Values y for record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).

Original entry on oeis.org

181, 3787, 174692, 685700, 2178889, 5931641, 31622776, 64631634, 1691869691, 2597429617, 16328969210, 22469029417, 54353589638, 380636413501, 2506650894908, 11290681881873, 12924394402851, 127673846293724

Offset: 1

Artur Jasinski, Jul 28 2010

nonn

Distance d is equal to 0 when x = k^2 and y = k^15.
For d values see A179812.
For x values see A179813.
Conjecture: For any positive number x >= A179813(n), the distance d between the fifteenth power of x and the square of any y (such that x <> k^2 and y <> k^15) can't be less than A179812(n).

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800, A179812, A179813, A179814.

d = 15; 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

A179447 Smallest values d such that the equation d =x^5-y^2 has exactly n distinct nonnegative integer solutions.

Original entry on oeis.org

2, 1, 7, 1044976, 11331151

Offset: 0

Artur Jasinski, Jul 14 2010

a(0)=2 because no integer solutions x^5-y^2 = 2;
a(1)=1 because 1=1^5-0^2;
a(2)=7 because 7=2^5-5^2 and 7=8^5-181^2;
a(3)=1044976 because 1044976=16^5-60^2 and 1044976=20^5-1468^2 and 1044976=41^5-10715^2;
a(4)=11331151 because 11331151=35^5-6418^2 and 11331151=40^5-9543^2 and 11331151=56^5-23225^2 and 11331151=386^5-2927305^2.

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, A179408, A179439.

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

Original entry on oeis.org

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

A179448 Numbers d such that the equation d =x^5-y^2 has more than 2 distinct nonnegative integer solutions.

Original entry on oeis.org

1044976, 1541468, 11331151, 15579791, 16410368, 33543196, 46539324, 72697500, 302272796, 528292607

Offset: 1

Artur Jasinski, Jul 14 2010

			a(1)=1044976 because 1044976=16^5-60^2 and 1044976=20^5-1468^2 and 1044976=41^5-10715^2;
a(3)=11331151 because 11331151=35^5-6418^2 and 11331151=40^5-9543^2 and 11331151=56^5-23225^2 and 11331151=386^5-2927305^2.

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, A179408, A179439, A179447.

cp's OEIS Frontend

A179406 Record minima of the positive distance d between the fifth power of a positive integer x and the square of an integer y such that d = x^5 - y^2 (x != k^2 and y != k^5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A179798 Record minima of the positive distance d between the 11th power of a positive integer x and the square of an integer y such that d = x^13 - y^2 (x <> k^2 and y <> k^13).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A179799 Values x for records of minima of the positive distance d between an 11th power of a positive integer x and a square of an integer y such that d = x^13 - y^2 (x<>k^2 and y<>k^13).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A179800 Values y for record minima of the positive distance d between the thirteenth power of a positive integer x and the square of an integer y such that d = x^13 - y^2 (x <> k^2 and y <> k^13).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A179812 Record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A179813 Values x for record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A179814 Values y for record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A179447 Smallest values d such that the equation d =x^5-y^2 has exactly n distinct nonnegative integer solutions.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A179448 Numbers d such that the equation d =x^5-y^2 has more than 2 distinct nonnegative integer solutions.

Views

Author

Keywords

Examples

Links

Crossrefs