勾股数新公式

蔡家雄

求解方程：\(y^2=8*x^2+1\) 的正整数解，

Treenewbee

蔡家雄 发表于 2023-12-2 13:08
求解方程：\(y^2=8*x^2+1\) 的正整数解，

\[x_n=\frac{\left(\sqrt{2}+1\right)^{2n}-\left(\sqrt{2}-1\right)^{2n}}{4 \sqrt{2}}\]
\[y_n=\frac{\left(\sqrt{2}+1\right)^{2n}+\left(\sqrt{2}-1\right)^{2n}}{2}\]

王守恩

Treenewbee 发表于 2023-11-30 08:16
1        {1,6}
2        {2,13}
3        {3,20}

谢谢 Treenewbee！还得再来一题。
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a/b}{\sqrt[n]{\sin(\pi/5)}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的{a,b}是怎样的一些数对？

Treenewbee

\[ n-1<n- \frac{a}{b}<=n\]
\[0<=\frac{a}{b}<1\]

王守恩

Treenewbee 发表于 2023-12-3 02:29
\[ n-1

谢谢 Treenewbee！
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a/b}{\sqrt[n]{\sin(\pi/5)}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的{a,b}是怎样的一些数对？
a=1, b=2--1,
a=2, b=3--3,
a=3, b=4--5,
a=4, b=5--7,
a=5, b=6--9,
a=6, b=7--11,
a=7, b=8--13,
a=8, b=9--15,
Table[{a, a + 1, -Ceiling[a/Log@Sin[\[Pi]/5]]}, {a, 30}]
{{1, 2, 1}, {2, 3, 3}, {3, 4, 5}, {4, 5, 7}, {5, 6, 9}, {6, 7, 11}, {7, 8, 13}, {8, 9, 15}, {9, 10, 16},
{10, 11, 18}, {11, 12, 20}, {12, 13, 22}, {13, 14, 24}, {14, 15, 26}, {15, 16, 28}, {16, 17, 30},
{17, 18, 31}, {18, 19, 33}, {19, 20, 35}, {20, 21, 37}, {21, 22, 39}, {22, 23, 41}, {23, 24, 43},
{24, 25, 45}, {25, 26, 47}, {26, 27, 48}, {27, 28, 50}, {28, 29, 52}, {29, 30, 54}, {30, 31, 56},

蔡家雄

求解方程：\(y^2=40*x^2+9\) 的正整数解，

Treenewbee

蔡家雄 发表于 2023-12-3 13:09
求解方程：\(y^2=40*x^2+9\) 的正整数解，

暂未找到通项公式。

1. Values@Solve[y^2==40x^2+9&&0<x<y<10^20,{x,y},Integers]

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{{1,7},{2,13},{9,57},{40,253},{77,487},{342,2163},{1519,9607},{2924,18493},{12987,82137},{57682,364813},{111035,702247},{493164,3119043},{2190397,13853287},{4216406,26666893},{18727245,118441497},{83177404,526060093},{160112393,1012639687},{711142146,4497657843},{3158550955,19976430247},{6080054528,38453641213},{27004674303,170792556537},{119941758886,758578289293},{230881959671,1460225726407},{1025466481368,6485619490563},{4554628286713,28805998562887},{8767434412970,55450123962253},{38940721617681,246282748084857},{172955933136208,1093869367100413},{332931625733189,2105644484839207},{1478721954990510,9352258807734003},{6567770830889191,41538229951252807},{12642634343448212,79959040299927613},{56152493568021699,355139551945807257},{249402335640653050,1577358868780506253},{480087173425298867,3036337886912410087},{2132316033629834052,13485950715132941763},{9470720983513926709,59898098783707984807}}

Treenewbee

Treenewbee 发表于 2023-12-3 14:25
暂未找到通项公式。

经过1个小时的演算，终于找到其通项公式，需要按模3分别处理：

\[x_n=\left(n-3 \left\lfloor \frac{n+2}{3}\right\rfloor \right) \cosh \left(2 \log \left(\sqrt{10}+3\right) \left\lceil \frac{n}{3}\right\rceil \right)+\frac{\left(23-4 \sqrt{19} (-1)^n \sin \left(\frac{\pi  n}{3}+\tan ^{-1}\left(\frac{7 \sqrt{3}}{9}\right)\right)\right) \sinh \left(2 \log \left(\sqrt{10}+3\right) \left\lceil \frac{n}{3}\right\rceil \right)}{6 \sqrt{10}}\]

1. x[n_]:=1/(6 Sqrt[10]) (23-4* (-1)^n *Sqrt[19] Sin[n *Pi/3+ArcTan[7Sqrt[3]/9]])Sinh[2Ceiling[n/3] *Log[3+Sqrt[10]]]+(n-3 Floor[(n+2)/3]) Cosh[2Ceiling[n/3] Log[3+ Sqrt[10]]];Table[x[n],{n,30}]//FullSimplify

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{1,2,9,40,77,342,1519,2924,12987,57682,111035,493164,2190397,4216406,18727245,83177404,160112393,711142146,3158550955,6080054528,27004674303,119941758886,230881959671,1025466481368,4554628286713,8767434412970,38940721617681,172955933136208,332931625733189,1478721954990510}

Treenewbee

1. y[n_]:=1/3 (23- (-1)^n(4 Sqrt[19] Sin[(n Pi )/3+ArcTan[7Sqrt[3]/9]]))Cosh[2Ceiling[n/3] *Log[3+Sqrt[10]]]+2*(n-3 Floor[(n+2)/3])Sqrt[10]Sinh[2Ceiling[n/3] Log[3+ Sqrt[10]]];Table[y[n],{n,12}]//FullSimplify

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{7,13,57,253,487,2163,9607,18493,82137,364813,702247,3119043}

\[y_n=2 \sqrt{10} \left(n-3 \left\lfloor \frac{n+2}{3}\right\rfloor \right) \sinh \left(2 \log \left(\sqrt{10}+3\right) \left\lceil \frac{n}{3}\right\rceil \right)+\frac{1}{3} \left(23-(-1)^n \left(4 \sqrt{19} \sin \left(\frac{\pi  n}{3}+\tan ^{-1}\left(\frac{7 \sqrt{3}}{9}\right)\right)\right)\right) \cosh \left(2 \log \left(\sqrt{10}+3\right) \left\lceil \frac{n}{3}\right\rceil \right)\]

王守恩

Treenewbee 发表于 2023-12-3 14:25
暂未找到通项公式。

我还是喜欢用这两个公式(不烧脑细胞)。

{0, 1, 2, 9, 40, 77, 342, 1519, 2924, 12987, 57682, 111035, 493164, 2190397, 4216406, 18727245,
83177404, 160112393, 711142146, 3158550955, 6080054528, 27004674303, 119941758886,

1. LinearRecurrence[{0, 0, 38, 0, 0, -1}, {0, 1, 2, 9, 40, 77}, 27]

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1. CoefficientList[Series[(0+x+2x^2+9x^3+2x^4+x^5)/(x^6-38x^3+1),{x,0,26}],x]

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{3, 7, 13, 57, 253, 487, 2163, 9607, 18493, 82137, 364813, 702247, 3119043, 13853287, 26666893,
118441497, 526060093, 1012639687, 4497657843, 19976430247, 38453641213, 170792556537,

1. LinearRecurrence[{0, 0, 38, 0, 0, -1}, {3, 7, 13, 57, 253, 487}, 26]

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1. CoefficientList[Series[(3+7x+13x^2-57x^3-13x^4-7x^5)/(x^6-38x^3+1),{x,0,9}],x]

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