A179406 -id:A179406 - cp's OEIS frontend

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).

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).

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

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.

A198393 Rank of hyperelliptic curve y^2 = x^5 - n.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 0, 2, 0, 0, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 2, 1, 3, 1, 2, 1, 0, 0, 1, 0, 2, 3, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1, 3, 1, 0, 1, 2, 0, 0, 1, 2, 1, 0, 1, 1, 1, 1

Offset: 1

Artur Jasinski, Oct 24 2011

nonn

If a(n)=0 number of rational points of hyperelliptic curve is finite and if a(n)<>0 then is infinite. For n when a(n)=0 see A198394.

Cf. A179406, A179407, A179408.

_ := PolynomialRing(Rationals());
for n := 1 to 100 do
C := HyperellipticCurve(x^5-n);
J := Jacobian(C);
RankBound(J)

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

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

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A198393 Rank of hyperelliptic curve y^2 = x^5 - n.

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

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