A179407 -id:A179407 - 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

A198443 Conjectured record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

Original entry on oeis.org

3, 4, 11, 26, 37, 368, 1828, 2180, 7825, 8177, 8217, 71393, 72481, 75154, 118409, 175485, 203697, 206370, 1049148, 1058224, 1843945, 1846618, 8186369, 8197633, 9600802, 96020524, 169503449, 294638801, 305158594, 305192969, 657099024

Offset: 1

Artur Jasinski, Oct 25 2011

Distance d is equal to 0 when x = k^2 and y = k^5.
Only the values of x < 10^8 have been searched/
For x values see A198444.
For y values see A198445.
Conjecture: For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of a positive integer x (such that x <> k^2 and y <> k^5) can't be less than A198443(n).

Cf. A179406, A179407, A179408, A198445.

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)] + 1; k = m^2 - n^5; 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, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]];  AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd

a(n) = (A198445(n))^2 - (A198444(n))^5.

A198444 Values x for record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

Original entry on oeis.org

1, 2, 5, 23, 27, 73, 96, 104, 396, 404, 432, 686, 723, 735, 1130, 1159, 2019, 2031, 3861, 5310, 18219, 18231, 25592, 25608, 44367, 200141, 213842, 308228, 390615, 390635, 549976, 631544, 1579129, 1657086, 2941211, 2941239, 5523608

Offset: 1

Artur Jasinski, Oct 25 2011

Distance d is equal to 0 when x = k^2 and y = k^5.
For d values see A198443.
For y values see A198445.
Conjecture: For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of x (such that x <> k^2 and y <> k^5) can't be less than A198443(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. A179406, A179407, A179408, A198443, A198445.

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)] + 1; k = m^2 - n^5; 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, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; vecx

A198445 Values y of record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

Original entry on oeis.org

2, 6, 56, 2537, 3788, 45531, 90298, 110302, 3120599, 3280601, 3878907, 12325663, 14055482, 14645977, 42923597, 45730778, 183164286, 185898039, 926295393, 2054642668, 44803437862, 44877249113, 104775699199, 104939539201, 414619915847, 17920089051165, 21146208937291, 52744869326263, 95361328242187, 9537353527343

Offset: 1

Artur Jasinski, Oct 25 2011

nonn

Distance d is equal to 0 when x = k^2 and y = k^5.
For d values see A198443.
For x values see A198444.
Conjecture: For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of x (such that x <> k^2 and y <> k^5) can't be less than A198443(n).

Cf. A179406, A179407, A179408, A198443, A198444.

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)] + 1; k = m^2 - n^5; 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, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; vecy

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).

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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).

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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).

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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).

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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).

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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).

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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).

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A198443 Conjectured record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

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A198444 Values x for record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

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A198445 Values y of record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

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