x^(n)+y^(n+1)+z^(n+2)=w^(n+3)

Treenewbee

王守恩 发表于 2023-3-5 08:44
尊敬的Treenewbee！请教一个问题。

\(题目：若x,y,d为正整数,\ \ x^2-dy^2=1, \ \ 则d=1, 4, 9, 16, 2 ...

佩尔方程x^2-dy^2=1有解的条件就是d非平方数

Treenewbee

1. dd = Table[n + Floor[1/2 + Sqrt[n]], {n, 1, 90}]
   
2. r = Table[{d,
   
3.    First[FindInstance[x^2 - d*y^2 == 1 && y > 0, {x, y},
   
4.      Integers]]}, {d, dd}]

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{{2,{x->-3,y->2}},{3,{x->-2,y->1}},{5,{x->-9,y->4}},{6,{x->-5,y->2}},{7,{x->-8,y->3}},{8,{x->-3,y->1}},{10,{x->-19,y->6}},{11,{x->-10,y->3}},{12,{x->-7,y->2}},{13,{x->-649,y->180}},{14,{x->-15,y->4}},{15,{x->-4,y->1}},{17,{x->-33,y->8}},{18,{x->-17,y->4}},{19,{x->-170,y->39}},{20,{x->-9,y->2}},{21,{x->-55,y->12}},{22,{x->-197,y->42}},{23,{x->-24,y->5}},{24,{x->-5,y->1}},{26,{x->-51,y->10}},{27,{x->-26,y->5}},{28,{x->-127,y->24}},{29,{x->-9801,y->1820}},{30,{x->-11,y->2}},{31,{x->-1520,y->273}},{32,{x->-17,y->3}},{33,{x->-23,y->4}},{34,{x->-35,y->6}},{35,{x->-6,y->1}},{37,{x->-73,y->12}},{38,{x->-37,y->6}},{39,{x->-25,y->4}},{40,{x->-19,y->3}},{41,{x->-2049,y->320}},{42,{x->-13,y->2}},{43,{x->-3482,y->531}},{44,{x->-199,y->30}},{45,{x->-161,y->24}},{46,{x->-24335,y->3588}},{47,{x->-48,y->7}},{48,{x->-7,y->1}},{50,{x->-99,y->14}},{51,{x->-50,y->7}},{52,{x->-649,y->90}},{53,{x->-66249,y->9100}},{54,{x->-485,y->66}},{55,{x->-89,y->12}},{56,{x->-15,y->2}},{57,{x->-151,y->20}},{58,{x->-19603,y->2574}},{59,{x->-530,y->69}},{60,{x->-31,y->4}},{61,{x->-1766319049,y->226153980}},{62,{x->-63,y->8}},{63,{x->-8,y->1}},{65,{x->-129,y->16}},{66,{x->-65,y->8}},{67,{x->-48842,y->5967}},{68,{x->-33,y->4}},{69,{x->-7775,y->936}},{70,{x->-251,y->30}},{71,{x->-3480,y->413}},{72,{x->-17,y->2}},{73,{x->-2281249,y->267000}},{74,{x->-3699,y->430}},{75,{x->-26,y->3}},{76,{x->-57799,y->6630}},{77,{x->-351,y->40}},{78,{x->-53,y->6}},{79,{x->-80,y->9}},{80,{x->-9,y->1}},{82,{x->-163,y->18}},{83,{x->-82,y->9}},{84,{x->-55,y->6}},{85,{x->-285769,y->30996}},{86,{x->-10405,y->1122}},{87,{x->-28,y->3}},{88,{x->-197,y->21}},{89,{x->-500001,y->53000}},{90,{x->-19,y->2}},{91,{x->-1574,y->165}},{92,{x->-1151,y->120}},{93,{x->-12151,y->1260}},{94,{x->-2143295,y->221064}},{95,{x->-39,y->4}},{96,{x->-49,y->5}},{97,{x->-62809633,y->6377352}},{98,{x->-99,y->10}},{99,{x->-10,y->1}}}

Treenewbee

A033313                Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.

Treenewbee

OEIS的代码：

1. Table[If[! IntegerQ[Sqrt[k]], {k,
   
2.     FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y},
   
3.      Integers]}, Nothing], {k, 2, 90}][[All, 2, 1, 1, 2]]

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{3,2,9,5,8,3,19,10,7,649,15,4,33,17,170,9,55,197,24,5,51,26,127,9801,11,1520,17,23,35,6,73,37,25,19,2049,13,3482,199,161,24335,48,7,99,50,649,66249,485,89,15,151,19603,530,31,1766319049,63,8,129,65,48842,33,7775,251,3480,17,2281249,3699,26,57799,351,53,80,9,163,82,55,285769,10405,28,197,500001,19}

Treenewbee

A033317                Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

王守恩

Treenewbee 发表于 2023-3-5 09:20
A033317                Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.

非常感谢 Treenewbee！

部分d未取到是因为x取值不夠大，比如d=13,{x->649,y->180};d=61,{x->1766319049,y->226153980}

x 的取值还不能照搬上次的方法。x,y太大了！上次的方法用不了。谢谢 Treenewbee！

只能分成2道题来做。

Table[Round[n + Sqrt[n]], {n, 1, 90}]
Table[FindInstance[{x^2 - d*y^2 == 1, x > 0, y > 0}, {y, x}, Integers,1], {d, 1, 99}]

Treenewbee

A084888        她的方法下载不了，可以有我们的方法吗？
---------------------------------------------------------------------------

1. Table [Length[
   
2.   PowersRepresentations[n^3, 2, 2] /. {0, _} -> Nothing], {n, 0, 100}]

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Treenewbee

{0,0,1,0,0,2,0,0,1,0,2,0,0,2,0,0,0,2,1,0,2,0,0,0,0,3,2,0,0,2,0,0,1,0,2,0,0,2,0,0,2,2,0,0,0,2,0,0,0,0,4,0,2,2,0,0,0,0,2,0,0,2,0,0,0,8,0,0,2,0,0,0,1,2,2,0,0,0,0,0,2,0,2,0,0,8,0,0,0,2,2,0,0,0,0,0,0,2,1,0,3}

Treenewbee

A084888        她的方法下载不了，可以有我们的方法吗？
---------------------------------------------------------------------------


或者：

1. Table[Length@
   
2.   Cases[PowersRepresentations[n^3, 2,
   
3.     2], {_?Positive, _?Positive}], {n, 0, 100}]

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王守恩

Treenewbee 发表于 2023-3-5 11:23
{0,0,1,0,0,2,0,0,1,0,2,0,0,2,0,0,0,2,1,0,2,0,0,0,0,3,2,0,0,2,0,0,1,0,2,0,0,2,0,0,2,2,0,0,0,2,0,0,0,0 ...

太伟大了！！！就这么简单！！谢谢Treenewbee !

七边形数 m*(5m -3)/2 表为两个不同的三角数之和((x (x + 1))/2 + (y (y + 1))/2)的通解公式。

n(3,9) 边形数 m*(5m -3)/2 表为两个不同的三角数之和((x (x + 1))/2 + (y (y + 1))/2)的通解公式。

Ceiling[FractionalPart[(Table[Dimensions[FindInstance[
{m/2 ((m (2 n - 4))/(3 + Cos[n \[Pi]]) - (2 n - 8)/(3 + Cos[n \[Pi]]))
== (x (x + 1))/2 + (y (y + 1))/2, y >= x > 0}, {x, y}, Integers, 1]]
// Total, {n, 3, 9}, {m, 1, 19}] + 1)^-1]]

{{0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0},
{0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1},
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0},
{0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0},
{0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0}}