整数三角形

王守恩

zeroieme 发表于 2020-12-3 06:53
$a=m_1 n_1 \left(m_2^2+n_2^2\right)$
$b=m_2 n_2 \left(m_1^2+n_1^2\right)$
$c=|m_1 n_2-m_2 n_1| ...

贪心的问：钝角三角形可以有解吗？
Solve[{ArcSin[a/k] + ArcSin[b/ k] + ArcSin[c/k] == \[Pi],
a == x y (z^2 + w^2), b == z w (x^2 + y^2), c == ( x w - y z) (x z + y w)}, {k, a, b, c}]

zeroieme

northwolves 发表于 2020-11-30 12:31
若使$d=\frac{2abc}{\sqrt{(a+b+c)(c+a-b)(b+c-a)(a+b-c)}}$为整数，abc有没有通解？

设外心在(0,0)，三条外心-顶点相连的半径，与X轴正向的夹角分别是 $0$、$4 \tan ^{-1}\left(\frac{m_1}{n_1}\right)$、$4 \tan ^{-1}\left(\frac{m_2}{n_2}\right)$。以此顶点坐标代入就得到有理通解，乘上公分母即可。



https://bbs.emath.ac.cn/thread-16169-1-1.html

王守恩

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}

王守恩

王守恩 发表于 2020-12-13 08:48
整理一下(轻装上阵)。

1，\(\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=1\)

谢谢各位大侠的宠爱！换个话题(13楼已经换了话题)！
试证：若\(a, b\ \)是正整数。则\(\ \sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}=\frac{1}{2}\ \)无解。

王守恩

王守恩 发表于 2020-12-14 09:28
谢谢各位大侠的宠爱！换个话题(13楼已经换了话题)！
试证：若\(a, b\ \)是正整数。则\(\ \sqrt[3]{a+\ ...

往前走！若\(a, b, c\ \)是正整数。
试证：\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=\frac{n}{m}\ \)
\(n=1, 2, 3, 4, 5, .....\ \ m=2, 3, 4, 5, .....\)
只要\(n, m\)是互质数，则\(\frac{n}{m}\)无解。

lsr314

$t=(a+sqrt(b))^(1/3)+(a-sqrt(b))^(1/3)$满足$t^9-6at^6+(27b-15a^2)t^3-8a^3=0$
由于t的最高次项系数是1，如果a和b都是正整数，t是有理数则必然也是有理整数。

王守恩

lsr314 发表于 2020-12-15 14:41
$t=(a+sqrt(b))^(1/3)+(a-sqrt(b))^(1/3)$满足$t^9-6at^6+(27b-15a^2)t^3-8a^3=0$
由于t的最高次项系数是1 ...

能举个例子吗？

      1，2，3 有反例？

1，若\(a, b, c\ \)是正整数。\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=\frac{1}{2}\ \)无解。

2，若\(a, b, c\ \)是正整数。\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=\frac{n}{m}\ \)
\(n=1, 2, 3, 4, 5, .....\ \ m=2, 3, 4, 5, .....\)
只要\(n, m\)是互质数，则\(\frac{n}{m}\)无解。

3，若\(a, b, c, k\ \)是正整数。\(\sqrt[k]{a+b\sqrt{c}}+\sqrt[k]{a-b\sqrt{c}}=\frac{n}{m}\ \)
\(n=1, 2, 3, 4, 5, .....\ \ m=2, 3, 4, 5, .....\)
只要\(n, m\)是互质数，则\(\frac{n}{m}\)无解。

lsr314

王守恩 发表于 2020-12-16 11:27
能举个例子吗？

      1，2，3 有反例？

更直接的方法是，代数整数的和、乘积、开方，以及它们的组合，结果仍然是代数整数，所以左边一定是代数整数，不可能是分母大于1的既约分数

王守恩

lsr314 发表于 2020-12-16 12:57
更直接的方法是，代数整数的和、乘积、开方，以及它们的组合，结果仍然是代数整数，所以左边一定是代数整 ...

通解公式，往下怎么走？

1，\(\sqrt[3]{a+b\sqrt{c}}+\sqrt[3]{a-b\sqrt{c}}=2\ \)通解公式
LinearRecurrence[{2, -1}, {1, 4}, 22]
{a=1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64}
LinearRecurrence[{2, -1}, {3, 4}, 22]
{b=3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}
LinearRecurrence[{2, -1}, {0, 1}, 22]
{c=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}

2，\(\sqrt[5]{a+b\sqrt{c}}+\sqrt[5]{a-b\sqrt{c}}=2\ \)通解公式
LinearRecurrence[{3, -3, 1}, {1, 16, 41}, 17]
{a=1, 16, 41, 76, 121, 176, 241, 316, 401, 496, 601, 716, 841, 976, 1121, 1276, 1441}
LinearRecurrence[{3, -3, 1}, {5, 16, 29}, 17]
{b=5, 16, 29, 44, 61, 80, 101, 124, 149, 176, 205, 236, 269, 304, 341, 380, 421}
LinearRecurrence[{3, -3, 1}, {0, 1, 2}, 17]
{c=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

3，\(\sqrt[7]{a+b\sqrt{c}}+\sqrt[7]{a-b\sqrt{c}}=2\ \)通解公式
LinearRecurrence[{4, -6, 4, -1}, {1, 64, 239, 568}, 14]
{a=1, 64, 239, 568, 1093, 1856, 2899, 4264, 5993, 8128, 10711, 13784, 17389, 21568}
LinearRecurrence[{4, -6, 4, -1}, {7, 64, 169, 328}, 14]
{b=7, 64, 169, 328, 547, 832, 1189, 1624, 2143, 2752, 3457, 4264, 5179, 6208}
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 2, 3}, 14]
{c=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

4，\(\sqrt[9]{a+b\sqrt{c}}+\sqrt[9]{a-b\sqrt{c}}=2\ \)通解公式
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 256, 1393, 4240, 9841}, 12]
{a=1, 256, 1393, 4240, 9841, 19456, 34561, 56848, 88225, 130816, 186961, 259216}
LinearRecurrence[{5, -10, 10, -5, 1}, {9, 256, 985, 2448, 4921}, 12]
{b=9, 256, 985, 2448, 4921, 8704, 14121, 21520, 31273, 43776, 59449, 78736}
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 2, 3, 4}, 12]
{c=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

lsr314

不过是把$((1+\sqrt(a))^n)^(1/n)+((1-\sqrt(a))^n)^(1/n)=2$展开而已