n=2024有解吗？

王守恩

\((1+\frac1a)(1+\frac1b) =1+\frac1x,(1+\frac1a)(1+\frac1c)=1+\frac1y,(1+\frac1b)(1+\frac1c)=1+\frac1z,(1+\frac1a)(1+\frac1b)(1+\frac1c)=1+\frac1n\)

  a, b, c, x, y, z, n 都是正整数。 a < b < c。譬如。n = 84。

\((1+\frac1{119})(1+\frac1{560}) =1+\frac1{98},(1+\frac1{119})(1+\frac1{594})=1+\frac1{99},(1+\frac1{560})(1+\frac1{594})=1+\frac1{288},(1+\frac1{119})(1+\frac1{560})(1+\frac1{594})=1+\frac1{84}\)

n = 2024有解吗？谢谢！！！

northwolves

n=2024无解。

n<=10000的解有137组(格式{a,b,c,x,y,z,n}）：

1        {20,84,119,16,17,49,14}
2        {35,90,104,25,26,48,20}
3        {54,440,539,48,49,242,44}
4        {119,560,594,98,99,288,84}
5        {104,1260,1455,96,97,675,90}
6        {170,2736,3059,160,161,1444,152}
7        {440,539,1539,242,342,399,209}
8        {279,1890,1952,243,244,960,216}
9        {390,935,1104,275,288,506,220}
10        {560,594,935,288,350,363,220}
11        {252,5060,5543,240,241,2645,230}
12        {350,8424,9099,336,337,4374,324}
13        {464,13020,13919,448,449,6727,434}
14        {560,3135,5984,475,512,2057,440}
15        {539,4752,4850,484,485,2400,440}
16        {740,1664,7695,512,675,1368,480}
17        {696,2870,4059,560,594,1681,492}
18        {594,19040,20195,576,577,9800,560}
19        {1456,2015,2820,845,960,1175,650}
20        {740,26676,28119,720,721,13689,702}
21        {1332,3440,3999,960,999,1849,774}
22        {923,10010,10152,845,846,5040,780}
23        {1785,2736,3059,1080,1127,1444,798}
24        {1599,2664,4550,999,1183,1680,819}
25        {902,36120,37883,880,881,18490,860}
26        {1325,4080,9009,1000,1155,2808,900}
27        {1935,3267,4256,1215,1330,1848,945}
28        {1080,47564,49679,1056,1057,24299,1034}
29        {1701,4255,6992,1215,1368,2645,1035}
30        {2090,4674,5015,1444,1475,2419,1121}
31        {2184,4094,7475,1424,1690,2645,1196}
32        {2184,4255,6992,1443,1664,2645,1196}
33        {1274,61200,63699,1248,1249,31212,1224}
34        {1455,14840,25704,1325,1377,9408,1260}
35        {1455,18720,18914,1350,1351,9408,1260}
36        {1484,77220,80135,1456,1457,39325,1430}
37        {3450,4599,5474,1971,2116,2499,1449}
38        {2375,7524,10449,1805,1935,4374,1539}
39        {2925,5719,7524,1935,2106,3249,1539}
40        {2204,7105,30855,1682,2057,5775,1595}
41        {3135,5984,7105,2057,2175,3248,1595}
42        {2639,4640,47124,1682,2499,4224,1624}
43        {2279,9804,14060,1849,1961,5776,1634}
44        {1710,95816,99179,1680,1681,48734,1652}
45        {2450,7524,77615,1848,2375,6859,1805}
46        {1952,117180,121023,1920,1921,59535,1890}
47        {2159,32130,32384,2023,2024,16128,1904}
48        {3380,8280,9359,2400,2483,4393,1910}
49        {3471,4991,128960,2047,3380,4805,2015}
50        {4185,6695,9269,2575,2883,3887,2015}
51        {2639,12375,47124,2175,2499,9801,2079}
52        {2639,18095,21384,2303,2349,9801,2079}
53        {3024,12375,14399,2430,2499,6655,2079}
54        {2210,141504,145859,2176,2177,71824,2144}
55        {3905,9300,11004,2750,2882,5040,2200}
56        {3485,11151,16932,2655,2890,6723,2295}
57        {4080,9009,13727,2808,3145,5439,2331}
58        {2484,168980,173879,2448,2449,85697,2414}
59        {4899,5750,40824,2645,4374,5040,2484}
60        {4899,9016,11430,3174,3429,5040,2484}
61        {3819,15275,21774,3055,3249,8977,2679}
62        {2774,199800,205275,2736,2737,101250,2700}
63        {3059,51680,52002,2888,2889,25920,2736}
64        {3080,234156,240239,3040,3041,118579,3002}
65        {4199,14364,52325,3249,3887,11270,3059}
66        {6083,8294,25200,3509,4900,6240,3080}
67        {6720,8294,18095,3712,4900,5687,3080}
68        {5775,7524,77615,3267,5375,6859,3135}
69        {6399,11375,13824,4095,4374,6240,3159}
70        {6902,7020,37583,3480,5831,5915,3185}
71        {3402,272240,278963,3360,3361,137780,3320}
72        {6324,10879,29325,3999,5202,7935,3519}
73        {3740,314244,321639,3696,3697,158949,3654}
74        {4179,79002,79400,3969,3970,39600,3780}
75        {7525,15264,18815,5040,5375,8427,3975}
76        {4094,360360,368459,4048,4049,182182,4004}
77        {6460,20519,22932,4913,5040,10829,4046}
78        {5984,20349,45695,4624,5291,14079,4199}
79        {5719,26960,43472,4718,5054,16640,4256}
80        {5340,43164,52865,4752,4850,23762,4360}
81        {4464,410780,419615,4416,4417,207575,4370}
82        {6278,33579,35259,5289,5329,17199,4599}
83        {10074,11315,33579,5329,7749,8463,4599}
84        {5655,46864,57680,5046,5150,25856,4640}
85        {4850,465696,475299,4800,4801,235224,4752}
86        {8555,18879,31464,5887,6726,11799,4959}
87        {8463,19343,32798,5887,6727,12167,4991}
88        {7955,16575,78624,5375,7224,13689,5031}
89        {5543,115920,116402,5290,5291,58080,5060}
90        {6815,36800,43239,5750,5887,19880,5075}
91        {5252,525300,535703,5200,5201,265225,5150}
92        {6720,32759,94094,5576,6272,24299,5264}
93        {7504,34839,37050,6174,6240,17955,5292}
94        {6200,67734,79235,5680,5750,36517,5300}
95        {5670,589784,601019,5616,5617,297674,5564}
96        {6699,48575,591744,5887,6624,44890,5829}
97        {13376,14079,43680,6859,10240,10647,5928}
98        {11304,17765,44744,6908,9024,12716,5984}
99        {6104,659340,671439,6048,6049,332667,5994}
100        {6554,734160,747155,6496,6497,370300,6440}
101        {12284,15687,107484,6889,11024,13689,6474}
102        {12284,16575,78624,7055,10624,13689,6474}
103        {9344,36134,51975,7424,7920,21315,6496}
104        {7175,164450,165024,6875,6876,82368,6600}
105        {8855,30750,165024,6875,8404,25920,6600}
106        {8855,49104,54900,7502,7625,25920,6600}
107        {8855,49104,76383,7502,7935,29889,6831}
108        {7020,814436,828359,6960,6961,410669,6902}
109        {18655,22385,24804,10175,10647,11766,7215}
110        {12879,24380,55384,8427,10449,16928,7314}
111        {7502,900360,915243,7440,7441,453870,7380}
112        {13376,19551,119756,7942,12032,16807,7448}
113        {12649,27104,67275,8624,10647,19320,7644}
114        {9540,72239,87290,8427,8600,39527,7685}
115        {11627,24310,367080,7865,11270,22800,7700}
116        {8000,992124,1007999,7936,7937,499999,7874}
117        {10880,42159,91884,8648,9728,28899,7904}
118        {11780,46189,50064,9386,9536,24024,7904}
119        {15875,22224,58800,9260,12500,16128,8000}
120        {18980,24309,35112,10658,12320,14364,8176}
121        {14840,25704,66639,9408,12137,18549,8244}
122        {8514,1089920,1106819,8448,8449,549152,8384}
123        {9099,226800,227474,8748,8749,113568,8424}
124        {17423,22400,73359,9800,14079,17160,8645}
125        {17423,27378,45980,10647,12635,17160,8645}
126        {9044,1193940,1211895,8976,8977,601425,8910}
127        {13090,45474,79287,10164,11235,28899,9009}
128        {12375,57239,78624,10175,10692,33124,9009}
129        {21114,21320,73439,10608,16399,16523,9269}
130        {18495,20960,179469,9825,16767,18768,9315}
131        {14651,29600,208494,9800,13689,25920,9360}
132        {16575,29600,78624,10625,13689,21504,9360}
133        {14651,38961,77440,10647,12320,25920,9360}
134        {18018,29600,56979,11200,13689,19480,9360}
135        {9590,1304376,1323419,9520,9521,656914,9452}
136        {11799,65549,189980,9999,11109,48734,9499}
137        {12935,59500,118404,10625,11661,39600,9750}

王守恩

northwolves 发表于 2025-5-5 09:35
n=2024无解。

n

主帖太难！丢了！！换一道！！！

$(1+\frac3a)(1+\frac3b)(1+\frac3c)(1+\frac3d) (1+\frac3f) (1+\frac3g)=\frac{2028}{2025}$

a, b, c, d, f, g = 正整数。 a < b < c < d < f < g。求:  a + b + c + d + f + g 最小值。

王守恩

从简单想起。

$(\frac{a_{1}+1}{a_{1}})(\frac{a_{2}+1}{a_{2}})(\frac{a_{3}+1}{a_{3}})(\frac{a_{4}+1}{a_{4}})\cdots(\frac{a_{y}+1}{a_{y}})=\frac{n+1}{n}$

$a_{1},a_{2},a_{3},a_{4},\cdots,a_{y},n$ = 正整数。$a_{1}<a_{2}<a_{3}<a_{4}<\cdots<a_{y}$  求:$\ a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{y}$ 最小值。

则 $ a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{y}最小值=n*y^2+\frac{y(y-1)}{2}$

贪心一点。

$(\frac{a_{1}+x}{a_{1}})(\frac{a_{2}+x}{a_{2}})(\frac{a_{3}+x}{a_{3}})(\frac{a_{4}+x}{a_{4}})\cdots(\frac{a_{y}+x}{a_{y}})=\frac{n+x}{n}$

$a_{1},a_{2},a_{3},a_{4},\cdots,a_{y},n,x$ = 正整数。$a_{1}<a_{2}<a_{3}<a_{4}<\cdots<a_{y}$  求:$\ a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{y}$ 最小值。

若 $ a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{y}最小值=n*y^2+\frac{x*y(y-1)}{2}$

对 x 我们应该怎样限制？？？

王守恩

譬如：来一道题。还是可以吓唬许多人的。

$(\frac{a_{1}+7}{a_{1}})(\frac{a_{2}+7}{a_{2}})(\frac{a_{3}+7}{a_{3}})(\frac{a_{4}+7}{a_{4}})\cdots(\frac{a_{12}+7}{a_{12}})=\frac{372}{365}$

$a_{1},a_{2},a_{3},a_{4},\cdots,a_{12}$ =12个不同的正整数。 求:$\ a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{12}$ 最小值。