x^3+y^4=z^5 的正整数解

王守恩

还是可以找出一些反例来。

`2^3+1^6=3^2\\
2^6+2^9=24^2\\
18^3+3^6=9^4\\
18^3+3^6=3^8\\
63^2+6^6=15^4\\
7^9+7^{10}=686^3\\
18^4+3^{10}=405^2\\
1458^3+27^6=9^{10}`

`a,b,c` 可以是 3 个奇数吗？我的电脑动不了。

葡萄糖

\begin{gather*}
X^{n}+Y^{n+1}=Z^{n+2}\\
\\
\left\{
\begin{split}
X&=(c^2-b^2)^{(n+2)}b^{(n^2+2n-1)}c^{(n+1)^2}\\
Y&=(c^2-b^2)^{(n+1)}b^{(n-1)(n+2)}c^{n(n+1)}\\
Z&=(c^2-b^2)^{n}b^{(n-1)(n+1)}c^{(n^2+1)}
\end{split}
\right.
\end{gather*}

王守恩

葡萄糖 发表于 2023-1-23 18:00
\begin{gather*}
X^{n}+Y^{n+1}=Z^{n+2}\\
\\

谢谢 葡萄糖！谢谢宝贵的资料！

也可以用 6 楼的公式。\(X^{n}+Y^{n+1}=Z^{n+2}\)

\(\big((2^2-1)^2\big)^0+\big((2^2-1)^1\big)^1=\big(2*(2^2-1)^0\big)^2\)

\(\big((2^3-1)^3\big)^1+\big((2^3-1)^2\big)^2=\big(2*(2^3-1)^1\big)^3\)

\(\big((2^4-1)^4\big)^2+\big((2^4-1)^3\big)^3=\big(2*(2^4-1)^2\big)^4\)

\(\big((2^5-1)^5\big)^3+\big((2^5-1)^4\big)^4=\big(2*(2^5-1)^3\big)^5\)

\(\big((2^6-1)^6\big)^4+\big((2^6-1)^5\big)^5=\big(2*(2^6-1)^4\big)^6\)

\(\big((2^7-1)^7\big)^5+\big((2^7-1)^6\big)^6=\big(2*(2^7-1)^5\big)^7\)

\(\big((2^8-1)^8\big)^6+\big((2^8-1)^7\big)^7=\big(2*(2^8-1)^6\big)^8\)

\(\big((2^9-1)^9\big)^7+\big((2^9-1)^8\big)^8=\big(2*(2^9-1)^7\big)^9\)

补充内容 (2023-1-31 15:06):
2 可以换为 3, 4, 5, 6, 7, 8, 9, ...

王守恩

OEIS没有这串数，就不可以有这串数吗？

第01组:2+3=6, 2+6=3, 3+6=2,
第02组:2+4=6, 2+6=4, 4+6=2,
第03组:3+4=6, 3+6=4, 4+6=3,
第01组:2+4=8, 2+8=4, 4+8=2,
第02组:2+6=8, 2+8=6, 6+8=2,
第03组:3+6=8, 3+8=6, 6+8=3,
第04组:4+6=8, 4+8=6, 6+8=4,
第01组:2+6=9, 2+9=6, 6+9=2,
第02组:3+6=9, 3+9=6, 6+9=3,
第03组:4+6=9, 4+9=6, 6+9=4,
第04组:6+8=9, 6+9=8, 8+9=6,
第01组:2+4=10, 2+10=4, 4+10=2,
第02组:2+5=10, 2+10=5, 5+10=2,
第03组:2+6=10, 2+10=6, 6+10=2,
第04组:2+8=10, 2+10=8, 8+10=2,
第05组:3+6=10, 3+10=6, 6+10=3,
第06组:4+5=10, 4+10=5, 5+10=4,
第07组:4+6=10, 4+10=6, 6+10=4,
第08组:4+8=10, 4+10=5, 5+10=4,
第09组:5+6=10, 5+10=6, 6+10=5,
第10组:5+8=10, 5+10=8, 8+10=5,
第11组:6+8=10, 6+10=8, 8+10=6,
第12组:6+9=10, 6+10=9, 9+10=6,

a(06)=03, 23,24,34,
a(07)=00,
a(08)=04, 24,26,36,46,
a(09)=04, 26,36,46,68,
a(10)=12, 24,25,45,26,36,46,56,28,48,58,68,69,
a(11)=00,
a(12)=22, 23,24,34,26,36,46,28,38,48,68,29,39,49,69,89,20,30,40,50,60,80,90
a(13)=00,
a(14)=26, 24,26,36,46,27,47,67,28,48,68,78,69,20,40,50,60,70,80,22,32,42,62,72,82,92,02,
a(15)=28, 35,36,46,56,68,39,49,59,69,20,30,40,50,60,80,90,22,32,42,52,62,82,92,02,64,84,04,24,
a(16)=25, 24,34,26,46,28,48,68,20,40,60,80,22,32,42,62,82,02,24,44,64,74,84,04,24,05,
a(17)=00,
a(18)=57, 23,24,34,26,36,46,28,38,48,68,29,39,49,69,89,20,30,40,60,80,90,22,32,42,62,82,92,02,24,34,44,64,74,84,94,04,24,25,35,45,55,65,85,95,05,25,45,26,36,4,66,86,96,06,26,46,56,
a(19)=00
a(20)=60, 24,25,45,26,36,46,56,28,48,58,68,20,40,50,60,80,90,22,32,42,52,62,82,02,24,44,54,64,84,04,24,25,45,55,65,85,95,05,25,45,26,46,56,66,86,06,26,46,28,38,48,58,68,88,98,08,28,48,58,68,
a(21)=
a(22)=
a(23)=00

northwolves

王守恩 发表于 2023-1-25 13:35
OEIS没有这串数，就不可以有这串数吗？

第01组:2+3=6, 2+6=3, 3+6=2,

不知所云

王守恩

已知a,b,c，求x^a+y^b=z^c 的正整数解

  "在a,b,c 3 个数中，只要能找出一对(1个数与另2个数的积)互素数的“都有解，好像是对的。

  "在a,b,c 3 个数中，不能找出一对(1个数与另2个数的的积)互素数的”不一定有解，

我们把这些“不一定有解的a,b,c 3 个数”先罗列出来：记 a,b,c 3 数中最大数=n，可以有a(n)种组合。

a(06)=03, 23,24,34,
a(07)=00,
a(08)=04, 24,26,36,46,
a(09)=04, 26,36,46,68,
a(10)=12, 24,25,45,26,36,46,56,28,48,58,68,69,
a(11)=00,
a(12)=22, 23,24,34,26,36,46,28,38,48,68,29,39,49,69,89,20,30,40,50,60,80,90
a(13)=00,
a(14)=26, 24,26,36,46,27,47,67,28,48,68,78,69,20,40,50,60,70,80,22,32,42,62,72,82,92,02,
a(15)=28, 35,36,46,56,68,39,49,59,69,20,30,40,50,60,80,90,22,32,42,52,62,82,92,02,64,84,04,24,
a(16)=25, 24,34,26,46,28,48,68,20,40,60,80,22,32,42,62,82,02,24,44,64,74,84,04,24,05,
a(17)=00,

OEIS没有这串数: 3, 0, 4, 4, 12, 0, 22, 0, 26, 28, 25, 0, ......

王守恩

3, 0, 4, 4, 12, 0, 22, 0, 26, 27, 30, 0, 58, 0, 64, 55, 73, 0, 111, 33, 103, 70, 133, 0, 215, 0, 144, 135, 181, 125, 275, 0, 228, 189, 307, 0, 424,
0, 328, 322, 343, 0, 499, 107, 486, 327, 465, 0, 638, 285, 587, 408, 554, 0, 928, 0, 628, 597, 642, 388, 1057, 0, 790, 598, 1133, 0, 1169, 0, 901,
940, 992, 470, 1472, 0, 1300, 761, 1119, 0, 1789, 644, 1231, 950, 1413, 0, 2139, 629, 1456, 1087, 1477, 793, 2126, 0, 1867, 1397, 2074, ......

Table[\(\D\sum_{b=3}^{(n - 1)}\sum_{c=2}^{(b - 1)}\bigg\lfloor\frac{\lceil FractionalPart(GCD(n, b)^{-1})\rceil +\lceil FractionalPart(GCD(b, c)^{-1})\rceil +\lceil FractionalPart(GCD(c, n)^{-1})\rceil}{2}\bigg\rfloor\), {n, 6, 100}]

王守恩

已知a,b,c，求x^a+y^b=z^c 的正整数解

  我们把“肯定无解的”罗列出来：记 a,b,c 3 数中最大数=n，可以有a(n)种组合。

第1对:4+4=2, 2+4=4

第1对:6+6=2, 2+6=6
第2对:6+6=3, 3+6=6
第3对:6+6=4, 4+6=6

第1对:8+8=2, 2+8=8
第2对:8+8=4, 4+8=8
第3对:8+8=6, 6+8=8

第1对:9+9=3, 3+9=9
第2对:9+9=6, 6+9=9

第1对:10+10=2, 2+10=10
第2对:10+10=4, 4+10=10
第3对:10+10=5, 5+10=10
第4对:10+10=6, 6+10=10
第5对:10+10=8, 8+10=10

第1对:12+12=2, 2+12=12
第2对:12+12=3, 3+12=12
第3对:12+12=4, 4+12=12
第4对:12+12=6, 6+12=12
第5对:12+12=8, 8+12=12
第6组:12+12=9, 9+12=12
第7组:12+12=0, 0+12=12
......

得到这样一串数。

0, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 7, 0, 7, 6, 7, 0, 11, 0, 11, 8, 11, 0, 15, 4, 13, 8, 15, 0, 21, 0, 15, 12, 17, 10, 23, 0, 19, 14, 23, 0,
29, 0, 23, 20, 23, 0, 31, 6, 29, 18, 27, 0, 35, 14, 31, 20, 29, 0, 43, 0, 31, 26, 31, 16, 45, 0, 35, 24, 45, 0, 47, 0, 37, 34, 39, 16,
53, 0, 47, 26, 41, 0, 59, 20, 43, 30, 47, 0, 65, 18, 47, 32, 47, 22, 63, 0, 55, 38, 59, 0, 69, 0, 55, 56, 53, 0, 71, 0, 69, 38, 63, ......

Table[\(\D\sum_{k=2}^{(n - 1)}\bigg\lceil FractionalPart(GCD(n, k)^{-1}\bigg\rceil\), {n, 1, 200}]

A016035        还没有这个通项公式。

王守恩

已知a,b,c，求x^a+y^b=z^c 的正整数解

记 a,b,c 3 数中最大数=n，可以有a(n)种组合。

a( n ):总数=无解+模糊+有解
a(02): 001= 000+ 000 +001
a(03): 004= 000+ 000 +004
a(04): 011= 002+ 000 +009
a(05): 021= 000+ 000 +021
a(06): 034= 010+ 009 +015
a(07): 050= 000+ 000 +050
a(08): 069= 010+ 012 +047
a(09): 091= 008+ 012 +071
a(10): 116= 018+ 036 +062
a(11): 144= 000+ 000 +144
a(12): 175= 026+ 066 +083
a(13): 209= 000+ 000 +209
a(14): 246= 026+ 078 +142
a(15): 286= 024+ 081 +181
a(16): 329= 026+ 090 +213
a(17): 375= 000+ 000 +375
a(18): 424= 042+ 174 +208
a(19): 476= 000+ 000 +476
a(20): 531= 042+ 192 +297
........

总数是这样一串数: 1, 4, 11, 21, 34, 50, 69, 91, 116, 144, 175, 209, 246, 286, 329, 375, 424, 476, 531, 589, 650, 714, 781, 851, 924,
1000, 1079, 1161, 1246, 1334, 1425, 1519, 1616, 1716, 1819, 1925, 2034, 2146, 2261, 2379, 2500, 2624, 2751, 2881, 3014, 3150, 3289,
3431, 3576, 3724, 3875, 4029, 4186, 4346, 4509, 4675, 4844, 5016, 5191, 5369, 5550, 5734, 5921, 6111, 6304, 6500, 6699, 6901, .......

无解是这样一串数: 0, 0, 2, 0, 10, 0, 10, 8, 18, 0, 26, 0, 26, 24, 26, 0, 42, 0, 42, 32, 42, 0, 58, 16, 50, 32, 58, 0, 82, 0, 58, 48, 66, 40,
90, 0, 74, 56, 90, 0, 114, 0, 90, 80, 90, 0, 122, 24, 114, 72, 106, 0, 138, 56, 122, 80, 114, 0, 170, 0, 122, 104, 122, 64, 178, 0, 138, 96,
178, 0, 186, 0, 146, 136, 154, 64, 210, 0, 186, 104, 162, 0, 234, 80, 170, 120, 186, 0, 258, 72, 186, 128, 186, 88, 250, 0, 218, 152, .....

模糊是这样一串数: 0, 0, 0, 0, 9, 0, 12, 12, 36, 0, 66, 0, 78, 81, 90, 0, 174, 0, 192, 165, 219, 0, 333, 99, 309, 210, 399, 0, 645, 0, 432,
405, 543, 375, 825, 0, 684, 567, 921, 0, 1272, 0, 984, 966, 1029, 0, 1497, 321, 1458, 981, 1395, 0, 1914, 855, 1761, 1224, 1662, 0, 2784,
0, 1884, 1791, 1926, 1164, 3171, 0, 2370, 1794, 3399, 0, 3507, 0, 2703, 2820, 2976, 1410, 4416, 0, 3900, 2283, 3357, 0, 5367, 1932, ....

有解是这样一串数: 1, 4, 9, 21, 15, 50, 47, 71, 62, 144, 83, 209, 142, 181, 213, 375, 208, 476, 297, 392, 389, 714, 390, 736, 565, 758,
622, 1161, 519, 1334, 935, 1066, 1007, 1301, 904, 1925, 1276, 1523, 1250, 2379, 1114, 2624, 1677, 1835, 1895, 3150, 1670, 3086, 2004,
2671, 2374, 4029, 2134, 3435, 2626, 3371, 3068, 5016, 2237, 5369, 3544, 3839, 3873, 4883, 2955, 6500, 4191, 5011, 3529, 7314, .......

王守恩

已知  n=2, 3, 4, 5, 6, 7, 8, 9, ......

试证:  \(x^{n}+y^{n+1}+z^{n+2}=w^{n+3}\)  有 >1 的正整数解。