不定方程a^4±bc^4+am=ab的整数解

yangchuanju · 发表于 2024-7-30 18:24

本帖最后由 yangchuanju 于 2024-7-30 18:26 编辑

对不定方程2*a^2+2*bc^4+a*c^4-a-2ab=0，已得到26组c=3的整数解，整数解成对成双出现，互相对称：
a b c f
-3186 -3068 3 0
78 -3068 3 0
-1008 -896 3 0
72 -896 3 0
-810 -700 3 0
70 -700 3 0
-282 -188 3 0
54 -188 3 0
-216 -128 3 0
48 -128 3 0
-40 0 3 0
0 0 3 0
-18 4 3 0
180 400 3 0
162 404 3 0
202 404 3 0
114 532 3 0
378 532 3 0
108 592 3 0
444 592 3 0
92 1104 3 0
972 1104 3 0
90 1300 3 0
1170 1300 3 0
84 3472 3 0
3348 3472 3 0

yangchuanju · 发表于 2024-7-30 18:26

对不定方程2*a^2+2*bc^4+a*c^4-a-2ab=0，已得到8组c=7的整数解，整数解成对成双出现，互相对称：
a b c f
0 0 7 0
-1200 0 7 0
-2058 -396 7 0
462 -396 7 0
4802 12004 7 0
6002 12004 7 0
4340 12400 7 0
6860 12400 7 0

yangchuanju · 发表于 2024-7-30 18:40

以下a,b,c都是不定方程的整数解
c a b
0 0 任意整数
1 2 4
2 32 79
5 -312 0
6 -576 22
6 192 -146
9 -3280 0
11 -7320 0
13 -14280 0
15 -25312 0

yangchuanju · 发表于 2024-7-30 18:59

本帖最后由 yangchuanju 于 2024-7-30 19:01 编辑

太阳先生的整数解
序号 a b c
1 -1299497230788526 -1299497019051308 109
2 -283773943550649876 -283773941198291088 199
3 -2126461065140485465476 -2126461064936853467088 199
4 -20577735824498028 -20577733990251696 187

yangchuanju · 发表于 2024-7-30 21:10

对不定方程2*a^2+2*bc^4+a*c^4-a-2ab=0，已得到8组c=5的整数解，整数解成对成双出现，互相对称：
a b c
-4060 -3248 5
500 -3248 5
-312 0 5
0 0 5
1250 3124 5
1562 3124 5
750 6372 5
5310 6372 5

c=3,5,7时，分别找到多组整数解！

yangchuanju · 发表于 2024-7-31 05:29

对不定方程2*a^2+2*bc^4+a*c^4-a-2ab=0，已得到6组c=9的整数解，
但根据整数解成对成双、互相对称出现的原则，没有找到第7-8组整数解，b=-39888、-40000、-40112都不是：
a b c
-2916 112 9
-252 112 9
-3280 0 9
0 0 9
9720 40000 9
27000 40000 9

太阳 · 发表于 2024-7-31 09:39

对不定方程2*a^2+2*bc^4+a*c^4-a-2ab=0，c＝9
数学软件，wolfram Alpha，找到72 整数解

太阳 · 发表于 2024-7-31 13:22

a = -64560240, b = -64550400
a = -21515706, b = -21505868
a = -7167528, b = -7157696
a = -4960116, b = -4950288
a = -2384802, b = -2374988
a = -1648998, b = -1639196
a = -790560, b = -780800
a = -545292, b = -535568
a = -259146, b = -249548
a = -177390, b = -167900
a = -82008, b = -72896
a = -78732, b = -69648
a = -54756, b = -45968
a = -22962, b = -15308
a = -21870, b = -14300
a = -13878, b = -7196
a = -3280, b = 0
a = -2916, b = 112
a = -252, b = 112
a = 0, b = 0
a = 3402, b = -7196
a = 4290, b = -14300
a = 4374, b = -15308
a = 5508, b = -45968
a = 5804, b = -69648
a = 5832, b = -72896
a = 6210, b = -167900
a = 6318, b = -249548
a = 6444, b = -535568
a = 6480, b = -780800
a = 6522, b = -1639196
a = 6534, b = -2374988
a = 6548, b = -4950288
a = 6552, b = -7157696
a = 6558, b = -21505868
a = 6560, b = -64550400
a = 6562, b = 64583204
a = 6564, b = 21538672
a = 6570, b = 7190500
a = 6574, b = 4983092
a = 6588, b = 2407792
a = 6600, b = 1672000
a = 6642, b = 813604
a = 6678, b = 568372
a = 6804, b = 282352
a = 6912, b = 200704
a = 7290, b = 105700
a = 7318, b = 102452
a = 7614, b = 78772
a = 8748, b = 48112
a = 8832, b = 47104
a = 9720, b = 40000
a = 13122, b = 32804
a = 13374, b = 32692
a = 16038, b = 32692
a = 16402, b = 32804
a = 27000, b = 40000
a = 34992, b = 47104
a = 36084, b = 48112
a = 67878, b = 78772
a = 91854, b = 102452
a = 95130, b = 105700
a = 190512, b = 200704
a = 272268, b = 282352
a = 558414, b = 568372
a = 803682, b = 813604
a = 1662120, b = 1672000
a = 2397924, b = 2407792
a = 4973238, b = 4983092
a = 7180650, b = 7190500
a = 21528828, b = 21538672
a = 64573362, b = 64583204

太阳 · 发表于 2024-7-31 13:23

2*a^2+2*bc^4+a*c^4-a-2ab=0，c=9
整数解
a = -64560240, b = -64550400
a = -21515706, b = -21505868
a = -7167528, b = -7157696
a = -4960116, b = -4950288
a = -2384802, b = -2374988
a = -1648998, b = -1639196
a = -790560, b = -780800
a = -545292, b = -535568
a = -259146, b = -249548
a = -177390, b = -167900
a = -82008, b = -72896
a = -78732, b = -69648
a = -54756, b = -45968
a = -22962, b = -15308
a = -21870, b = -14300
a = -13878, b = -7196
a = -3280, b = 0
a = -2916, b = 112
a = -252, b = 112
a = 0, b = 0
a = 3402, b = -7196
a = 4290, b = -14300
a = 4374, b = -15308
a = 5508, b = -45968
a = 5804, b = -69648
a = 5832, b = -72896
a = 6210, b = -167900
a = 6318, b = -249548
a = 6444, b = -535568
a = 6480, b = -780800
a = 6522, b = -1639196
a = 6534, b = -2374988
a = 6548, b = -4950288
a = 6552, b = -7157696
a = 6558, b = -21505868
a = 6560, b = -64550400
a = 6562, b = 64583204
a = 6564, b = 21538672
a = 6570, b = 7190500
a = 6574, b = 4983092
a = 6588, b = 2407792
a = 6600, b = 1672000
a = 6642, b = 813604
a = 6678, b = 568372
a = 6804, b = 282352
a = 6912, b = 200704
a = 7290, b = 105700
a = 7318, b = 102452
a = 7614, b = 78772
a = 8748, b = 48112
a = 8832, b = 47104
a = 9720, b = 40000
a = 13122, b = 32804
a = 13374, b = 32692
a = 16038, b = 32692
a = 16402, b = 32804
a = 27000, b = 40000
a = 34992, b = 47104
a = 36084, b = 48112
a = 67878, b = 78772
a = 91854, b = 102452
a = 95130, b = 105700
a = 190512, b = 200704
a = 272268, b = 282352
a = 558414, b = 568372
a = 803682, b = 813604
a = 1662120, b = 1672000
a = 2397924, b = 2407792
a = 4973238, b = 4983092
a = 7180650, b = 7190500
a = 21528828, b = 21538672
a = 64573362, b = 64583204

yangchuanju · 发表于 2024-7-31 16:20

太阳先生的72组整数解重排序
序 a b b增原序和
1 -64560240 -64550400 ——
36 6560 -64550400 37
2 -21515706 -21505868 43044532
35 6558 -21505868 37
3 -7167528 -7157696 14348172
34 6552 -7157696 37
4 -4960116 -4950288 2207408
33 6548 -4950288 37
5 -2384802 -2374988 2575300
32 6534 -2374988 37
6 -1648998 -1639196 735792
31 6522 -1639196 37
7 -790560 -780800 858396
30 6480 -780800 37
8 -545292 -535568 245232
29 6444 -535568 37
9 -259146 -249548 286020
28 6318 -249548 37
10 -177390 -167900 81648
27 6210 -167900 37
11 -82008 -72896 95004
26 5832 -72896 37
12 -78732 -69648 3248
25 5804 -69648 37
13 -54756 -45968 23680
24 5508 -45968 37
14 -22962 -15308 30660
23 4374 -15308 37
15 -21870 -14300 1008
22 4290 -14300 37
16 -13878 -7196 7104
21 3402 -7196 37
17 -3280 0 7196
20 0 0 37
18 -2916 112 112
19 -252 112 37
54 13374 32692 32580
55 16038 32692 109
53 13122 32804 112
56 16402 32804 109
52 9720 40000 7196
57 27000 40000 109
51 8832 47104 7104
58 34992 47104 109
50 8748 48112 1008
59 36084 48112 109
49 7614 78772 30660
60 67878 78772 109
48 7318 102452 23680
61 91854 102452 109
47 7290 105700 3248
62 95130 105700 109
46 6912 200704 95004
63 190512 200704 109
45 6804 282352 81648
64 272268 282352 109
44 6678 568372 286020
65 558414 568372 109
43 6642 813604 245232
66 803682 813604 109
42 6600 1672000 858396
67 1662120 1672000 109
41 6588 2407792 735792
68 2397924 2407792 109
40 6574 4983092 2575300
69 4973238 4983092 109
39 6570 7190500 2207408
70 7180650 7190500 109
38 6564 21538672 14348172
71 21528828 21538672 109
37 6562 64583204 43044532
72 64573362 64583204 109

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不定方程a^4±bc^4+am=ab的整数解

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