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\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\)

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发表于 2020-12-10 08:42 | 显示全部楼层 |阅读模式
凑个热闹!
求\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\) 全部正整数解
发表于 2020-12-10 10:52 | 显示全部楼层
全部非平凡解是 \((a,b,c) = (3cn^2+1,n(cn^2+3),c)\).
其中\(c > 1\) 是任意无重因子的整数, \(n\) 是任意正整数.


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发表于 2020-12-10 10:56 | 显示全部楼层
例:\(c=6,\;n=5\) 时 \(\sqrt[3]{451+765\sqrt{6}}+\sqrt[3]{451-765\sqrt{6}}=2\)
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 楼主| 发表于 2020-12-10 13:38 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-11 15:45 编辑
elim 发表于 2020-12-10 10:52
全部非平凡解是 \((a,b,c) = (3cn^2+1,n(cn^2+3),c)\).
其中\(c > 1\) 是任意无重因子的整数, \(n\) 是任 ...


谢谢 elim!

\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\)
{01, 3, 0},{01, 06, 0},{001, 009, 0},{001, 012, 0},{001, 015, 0},
{04, 4, 1},{13, 14, 1},{028, 036, 1},{049, 076, 1},{076, 140, 1},
{07, 5, 2},{25, 22, 2},{055, 063, 2},{097, 140, 2},{151, 265, 2},
{10, 6, 3},{37, 30, 3},{082, 090, 3},{145, 204, 3},{226, 390, 3},
{13, 7, 4},{49, 38, 4},{109, 117, 4},{193, 268, 4},{301, 515, 4},
{16, 8, 5},{61, 46, 5},{136, 144, 5},{241, 332, 5},{376, 640, 5},
{19, 9, 6},{73, 54, 6},{163, 171, 6},{289, 396, 6},{451, 765, 6},
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 楼主| 发表于 2020-12-11 06:38 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-11 06:40 编辑

凑个热闹!
求\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\) 全部正整数解

贪心一点!
\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=n\ \ n=1,2,3,4,...\)都可以有通解公式
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 楼主| 发表于 2020-12-11 11:46 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-11 11:47 编辑
王守恩 发表于 2020-12-11 06:38
凑个热闹!
求\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\) 全部正整数解


给出“起点解”(还有比这更小的吗)?
\(\ \sqrt[3]{2+1\sqrt{5}}+\sqrt[3]{2-1\sqrt{5}}=1\)
\(\ \ \ \sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}=3\)
\(\ \sqrt[3]{25+10\sqrt{5}}+\sqrt[3]{25-10\sqrt{5}}=5\)
\(\ \ \ \sqrt[3]{56+19\sqrt{5}}+\sqrt[3]{56-19\sqrt{5}}=7\)
\(\ \sqrt[3]{108+31\sqrt{5}}+\sqrt[3]{108-31\sqrt{5}}=9\)
\(\ \sqrt[3]{187+46\sqrt{5}}+\sqrt[3]{187-46\sqrt{5}}=11\)
\(\ \sqrt[3]{299+64\sqrt{5}}+\sqrt[3]{299-64\sqrt{5}}=13\)
\(\ \ \sqrt[3]{450+85\sqrt{5}}+\sqrt[3]{450-85\sqrt{5}}=15\)
\(\ \sqrt[3]{646+109\sqrt{5}}+\sqrt[3]{646-109\sqrt{5}}=17\)
\(\ \ \sqrt[3]{893+136\sqrt{5}}+\sqrt[3]{893-136\sqrt{5}}=19\)
\(\ \sqrt[3]{1197+166\sqrt{5}}+\sqrt[3]{1197-166\sqrt{5}}=21\)
\(\ \sqrt[3]{1564+199\sqrt{5}}+\sqrt[3]{1564-199\sqrt{5}}=23\)
\(\ \sqrt[3]{2000+235\sqrt{5}}+\sqrt[3]{2000-235\sqrt{5}}=25\)
\(\ \sqrt[3]{2511+274\sqrt{5}}+\sqrt[3]{2511-274\sqrt{5}}=27\)
\(\ \sqrt[3]{3103+316\sqrt{5}}+\sqrt[3]{3103-316\sqrt{5}}=29\)
\(\ \sqrt[3]{3782+361\sqrt{5}}+\sqrt[3]{3782-361\sqrt{5}}=31\)
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 楼主| 发表于 2020-12-11 15:39 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-11 15:42 编辑

\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=1\)
{02, 1, 05},{017, 018, 05},{047, 080, 05},{092, 0217, 05},{0152, 0459, 05},
{05, 2, 13},{044, 045, 13},{122, 205, 13},{239, 0560, 13},{0395, 1188, 13},
{08, 3, 21},{071, 072, 21},{197, 330, 21},{386, 0903, 21},{0638, 1917, 21},
{11, 4, 29},{098, 099, 29},{272, 455, 29},{533, 1246, 29},{0881, 2646, 29},
{14, 5, 37},{125, 126, 37},{347, 580, 37},{680, 1589, 37},{1124, 3375, 37},
{17, 6, 45},{152, 153, 45},{422, 705, 45},{827, 1932, 45},{1367, 4104, 45},
{20, 7, 53},{179, 180, 53},{497, 830, 53},{974, 2275, 53},{1610, 4833, 53},

\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=3\)
{09, 04, 05},{054, 027, 05},{0144, 095, 05},{0279, 0238, 05},{0459, 0486, 05},
{18, 05, 13},{135, 054, 13},{0369, 220, 13},{0720, 0581, 13},{1188, 1215, 13},
{27, 06, 21},{216, 081, 21},{0594, 345, 21},{1161, 0924, 21},{1917, 1944, 21},
{36, 07, 29},{297, 108, 29},{0819, 470, 29},{1602, 1267, 29},{2646, 2673, 29},
{45, 08, 37},{378, 135, 37},{1044, 595, 37},{2043, 1610, 37},{3375, 3402, 37},
{54, 09, 45},{459, 162, 45},{1269, 720, 45},{2484, 1953, 45},{4104, 4131, 45},
{63, 10, 53},{540, 189, 53},{1494, 845, 53},{2925, 2275, 53},{4833, 4860, 53},

\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=5\)
{025, 10, 05},{100, 045, 05},{0250, 125, 05},{0475, 0280, 05},{0775, 0540, 05},
{040, 11, 13},{235, 072, 13},{0625, 250, 13},{1210, 0623, 13},{1990, 1269, 13},
{055, 12, 21},{370, 099, 21},{1000, 375, 21},{1945, 0966, 21},{3205, 1998, 21},
{070, 13, 29},{505, 126, 29},{1375, 500, 29},{2680, 1309, 29},{4420, 2727, 29},
{085, 14, 37},{640, 153, 37},{1750, 625, 37},{3415, 1652, 37},{5635, 3456, 37},
{100, 15, 45},{775, 180, 45},{2125, 750, 45},{4150, 1995, 45},{6850, 4185, 45},
{115, 16, 53},{910, 207, 53},{2500, 875, 53},{4885, 2338, 53},{8065, 4914, 53},

这可是我手工一个一个一个一个一个敲出来的,难免有错,高人们可有好办法?
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 楼主| 发表于 2020-12-11 16:02 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-11 16:04 编辑
王守恩 发表于 2020-12-11 15:39
\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=1\)
{02, 1, 05},{017, 018, 05},{047, 080, 05},{092 ...


凑个热闹!
求\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\) 全部正整数解

我可要换话题了!
\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=\frac{1}{n}\ \ n=2,3,4,...\)可有解?
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 楼主| 发表于 2020-12-12 20:30 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-12 20:32 编辑
王守恩 发表于 2020-12-11 16:02
凑个热闹!
求\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\) 全部正整数解


凑个热闹!

\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=\frac{1}{n}\ \ n=2,3,4,...\)可有正整数解?
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 楼主| 发表于 2020-12-13 08:51 | 显示全部楼层
本帖最后由 王守恩 于 2020-12-13 11:44 编辑

整理一下(轻装上阵)。

1,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=1\)
LinearRecurrence[{2, -1}, {2, 5}, 14]
{a=2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41}
LinearRecurrence[{4, -6, 4, -1}, {5, 52, 189, 464}, 14]
{b=5, 52, 189, 464, 925, 1620, 2597, 3904, 5589, 7700, 10285, 13392, 17069, 21364}

2,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=2\)
LinearRecurrence[{2, -1}, {1, 4}, 16]
{a=1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46}
LinearRecurrence[{4, -6, 4, -1}, {0, 16, 50, 108}, 16]
{b=0, 16, 50, 108, 196, 320, 486, 700, 968, 1296, 1690, 2156, 2700, 3328, 4046, 4860}

3,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=3\)
LinearRecurrence[{2, -1}, {9, 18}, 13]
{a=9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117}
LinearRecurrence[{4, -6, 4, -1}, {80, 325, 756, 1421}, 13]
{b=80, 325, 756, 1421, 2368, 3645, 5300, 7381, 9936, 13013, 16660, 20925, 25856}

4,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=4\)
LinearRecurrence[{2, -1}, {8, 14}, 14]
{a=8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86}
LinearRecurrence[{4, -6, 4, -1}, {0, 169, 392, 675}, 14]
{b=0, 169, 392, 675, 1024, 1445, 1944, 2527, 3200, 3969, 4840, 5819, 6912, 8125}

5,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=5\)
LinearRecurrence[{2, -1}, {25, 40}, 12]
{a=25, 40, 55, 70, 85, 100, 115, 130, 145, 160, 175, 190}
LinearRecurrence[{4, -6, 4, -1}, {500, 1573, 3024, 4901}, 12]
{b=500, 1573, 3024, 4901, 7252, 10125, 13568, 17629, 22356, 27797, 34000, 41013}

6,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=6\)
LinearRecurrence[{2, -1}, {27, 36}, 13]
{a=27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135}
LinearRecurrence[{4, -6, 4, -1}, {0, 784, 1682, 2700}, 13]
{b=0, 784, 1682, 2700, 3844, 5120, 6534, 8092, 9800, 11664, 13690, 15884, 18252}

7,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=7\)
LinearRecurrence[{2, -1}, {56, 77}, 11]
{a=56, 77, 98, 119, 140, 161, 182, 203, 224, 245, 266}
LinearRecurrence[{4, -6, 4, -1}, {1805, 5200, 9261, 14036}, 11]
{b=1805, 5200, 9261, 14036, 19573, 25920, 33125, 41236, 50301, 60368, 71485}

8,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=8\)
LinearRecurrence[{2, -1}, {64, 76}, 12]
{a=64, 76, 88, 100, 112, 124, 136, 148, 160, 172, 184, 196}
LinearRecurrence[{4, -6, 4, -1}, {0, 2401, 5000, 7803}, 12]
{b=0, 2401, 5000, 7803, 10816, 14045, 17496, 21175, 25088, 29241, 33640, 38291}

9,\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=9\)
LinearRecurrence[{2, -1}, {108, 135}, 11]
{a=108, 135, 162, 189, 216, 243, 270, 297, 324, 351, 378}
LinearRecurrence[{4, -6, 4, -1}, {4805, 13312, 22869, 33524}, 11]
{b=4805, 13312, 22869, 33524, 45325, 58320, 72557, 88084, 104949, 123200, 142885}
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