数论小猜想

蔡家雄

蔡氏完全循环节问题

若 \(2*10^n - 51\) 是素数，则 10 是这个素数的原根。

谢谢树新蜂老师提供100000以内的 n={2, 3, 4, 8, 11, 13, 17, 28, 56, 105, 231, 339, 643, 922, 1219, 1880, 2209, 4238, 4987, 14770, 56194, 67043, 96867}

wlc1

鲁氏求一解法，不是大衍求一术

恒等式：\(2^n+2^n=2^{n+1}\)

恒等式：\(2^n+2^n+2^{n+1}=2^{n+2}\)

求：\(x^{n+1}+y^{n+1}=z^{n}\)

解：\((2^{n-1})^{n+1}+(2^{n-1})^{n+1}=(2^{n})^{n}\)

求：\(x^n+y^{n+1}=z^n\)

解：\((a^n-1)^n+(a^n-1)^{n+1}=(a*(a^n-1))^n\)，\(a > 1\) .

求：\(x^{n+1}+y^n=z^{n+1}\)

解：\((a^{n^2-1}-1)^{n+1}+(a^{n^2-1}-1)^n=(a*(a^{n^2-1}-1))^{n+1}\)，\(a > 1\) .

求：\(x^{2n}+y^{2n+1}=z^{2n+2}\)

解：\(((a^{2n+2}-1)^{2n+2})^{2n}+((a^{2n+2}-1)^{2n+1})^{2n+1}=(a*(a^{2n+2}-1)^{2n})^{2n+2}\)，\(a > 1\) .

求：\(x^{2n+1}+y^{2n+2}=z^{2n+3}\)

解：\((2^{(n+1)(2n+2})^{2n+1}+(2^{(n+1)(2n+1})^{2n+2}=(2^{((n+1)(2n+1)(2n+2)+1)/(2n+3)})^{2n+3}\)

朱明君

恒等式：1^n+2^3=3^2
恒等式：a^2+b^2=c^2

朱明君

蔡家雄 发表于 2024-2-3 12:02
求：\(x^{2021}+y^{2023}=z^{2025}\)

解：\((2^{511819})^{2021}+(2^{511313})^{2023}=(2^{510808})^{ ...

\(\left( 2^{69}\right)^{21}+\left( 2^{63}\right)^{23}=\left( 2^{58}\right)^{25}\)

朱明君

，，，，，，

朱明君

\(设n为正整数，n\left( \frac{n+1}{8}\right)=m，\)
\(则\left( 2^m\right)^{n-2}+\left( 2^{m-\left( \frac{n+1}{4}\right)}\right)^n=\left( 2^{m-\left( \frac{n-1}{2}\right)}\right)^{n+2}\)
\(注：其中n是\left( 8N-1\right)的数都有解\)

lusishun

朱明君 发表于 2024-2-16 04:55
\(设n为正整数，n\left( \frac{n+1}{8}\right)=m，\)
\(则\left( 2^m\right)^{n-2}+\left( 2^{m-\left( \f ...

朱先生，
您解，X^63+y^71=z^79

Treenewbee

蔡家雄 发表于 2024-3-5 07:54
求：\(a^3+b^3+c^3+3abc=d^3\)

{{3, 10, 38, 39}, {6, 20, 76, 78}, {10, 26, 27, 39}, {20, 52, 54, 78}}

Treenewbee

蔡家雄 发表于 2024-4-27 17:30
求：1, 11, 109, 1079, ... 的通解公式，

\[a_n=\left\lfloor \frac{1}{4} \left(\sqrt{6}+2\right) \left(2 \sqrt{6}+5\right)^{n-1}\right\rfloor\]

王守恩

蔡家雄 发表于 2024-8-18 17:26
三角数A=n(n+1)/2的数字和=1，求A的通解公式，，，，

三角数A=n(n+1)/2的数字和=1，求A的通解公式，，，，
数字和=1,   A={1, 4}——只有这么2个？
数字和=2,   A={},
数字和=3,   A={2, 6, 15, 20, 24, 141, 200, 2000, 20000, 200000, 2000000,
数字和=4,   A={},
数字和=5,   A={},
数字和=6,   A={3, 5, 14, 21, 66, 77, 201, 473, 2001, 15620, 20001, 200001, 2000001,
数字和=7,   A={},
数字和=8,   A={},
数字和=9,   A={8, 9, 17, 18, 26, 35, 45, 53, 63, 80, 81, 89, 126, 144, 161, 162, 179, 206, 215, 224, 449, 458, 477, 666, 800, 801, 1421, 1575, 1620, 1673, 2006, 2015, 2195, 2835, 4473, 4733, 6326, 8000, 8001, 8126, 14184, 14220,  
数字和=10, A={7, 10, 13, 16, 19, 22, 25, 28, 37, 40, 46, 49, 55, 58, 64, 73, 79, 82, 100, 118, 142, 148, 157, 175, 184, 202, 205, 214, 220, 253, 286, 319, 322, 346, 379, 400, 424, 460, 469, 475, 637, 640, 775, 778, 802, 1000, 1414,
数字和=11, A= {},
数字和=12, A={11, 29, 33, 38, 42, 47, 51, 60, 65, 69, 78, 101, 110, 119, 146, 150, 155, 159, 164, 173, 195, 204, 245, 249, 258, 317, 326, 375, 402, 447, 510, 533, 600, 632, 663, 681, 722, 1001, 1095, 1122, 1266, 1415, 1424, 1428,
数字和=13, A={},
数字和=14, A={},
数字和=15, A={12, 23, 30, 32, 39, 41, 48, 50, 57, 59, 68, 75, 84, 86, 93, 95, 102, 104, 111, 113, 120, 122, 149, 156, 158, 167, 174, 185, 203, 210, 219, 221, 228, 237, 246, 255, 257, 266, 273, 284, 293, 300, 318, 320, 374, 401, 453,
数字和=16, A={},
数字和=17, A={},
数字和=18, A={27, 36, 44, 54, 62, 71, 72, 90, 98, 99, 117, 134, 135, 143, 152, 153, 170, 171, 180, 197, 198, 207, 216, 225, 233, 242, 251, 260, 261, 269, 279, 287, 288, 297, 323, 324, 333, 341, 350, 359, 377, 378, 395, 404, 405, 440,
数字和=19, A={31, 34, 43, 52, 61, 67, 70, 76, 85, 88, 91, 94, 97, 103, 106, 112, 115, 121, 124, 127, 130, 133, 136, 139, 145, 151, 154, 160, 163, 166, 169, 178, 181, 190, 193, 196, 199, 208, 211, 217, 226, 229, 232, 235, 238, 241, 247,
数字和=20, A={},
数字和=21, A={56, 74, 83, 87, 92, 96, 105, 114, 123, 128, 137, 168, 177, 182, 186, 191, 209, 213, 218, 222, 240, 254, 263, 272, 276, 281, 285, 290, 294, 299, 303, 321, 330, 335, 339, 344, 348, 353, 357, 362, 366, 371, 380, 384, 393,