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xbtianlang 发表于 2024-3-22 23:04:51

素阶乘猜想：p = P# - q

素阶乘猜想：对于大于5的素数p，存在素数P，q使得p = P# - q
其中P#为素数阶乘，P≥5，q>5

xbtianlang 发表于 2024-3-23 13:52:17

2024-03-23 13:49:22
7= 5# - {...}
11= 5# - {...} = 7# - {...}
13= 5# - {...} = 7# - {...} = 11# - {...}
17= 5# - {...} = 7# - {...} = 11# - {...} = 13# - {...}
19= 5# - {...} = 7# - {...} = 13# - {...}
23= 5# - {...} = 11# - {...} = 19# - {...}
29= 7# - {...} = 11# - {...} = 17# - {...}
31= 7# - {...}
37= 7# - {...} = 11# - {...} = 19# - {...}
41= 11# - {...} = 13# - {...} = 19# - {...} = 29# - {...}
43= 7# - {...} = 11# - {...} = 19# - {...} = 23# - {...}
47= 7# - {...} = 13# - {...} = 17# - {...} = 19# - {...} = 41# - {...}
53= 7# - {...} = 17# - {...} = 19# - {...} = 41# - {...}
59= 7# - {...} = 11# - {...} = 17# - {...} = 19# - {...} = 37# - {...}
61= 7# - {...} = 17# - {...} = 23# - {...}
67= 11# - {...} = 19# - {...} = 29# - {...} = 47# - {...}
71= 7# - {...} = 11# - {...} = 13# - {...} = 41# - {...}
73= 7# - {...} = 11# - {...} = 29# - {...} = 31# - {...} = 53# - {...}
79= 7# - {...} = 19# - {...} = 23# - {...} = 31# - {...}
83= 7# - {...} = 13# - {...} = 31# - {...} = 79# - {...}
89= 11# - {...} = 31# - {...} = 41# - {...} = 43# - {...} = 53# - {...} = 61# - {...} = 83# - {...}
97= 7# - {...} = 11# - {...} = 19# - {...} = 23# - {...} = 31# - {...} = 47# - {...} = 73# - {...} = 89# - {...}
101= 7# - {...} = 23# - {...} = 29# - {...} = 67# - {...} = 79# - {...} = 89# - {...}
103= 7# - {...} = 11# - {...} = 13# - {...} = 41# - {...} = 89# - {...}
107= 7# - {...} = 11# - {...} = 17# - {...} = 31# - {...} = 37# - {...} = 59# - {...} = 67# - {...}
109= 7# - {...} = 13# - {...} = 17# - {...} = 59# - {...}
113= 7# - {...} = 13# - {...} = 23# - {...} = 67# - {...} = 103# - {...}
127= 7# - {...} = 17# - {...} = 19# - {...} = 23# - {...} = 31# - {...} = 53# - {...} = 67# - {...} = 71# - {...} = 79# - {...}
131= 7# - {...} = 11# - {...} = 17# - {...} = 41# - {...} = 101# - {...}
137= 7# - {...} = 23# - {...} = 41# - {...} = 53# - {...} = 79# - {...} = 103# - {...}
139= 7# - {...} = 19# - {...} = 37# - {...} = 53# - {...} = 101# - {...}
149= 7# - {...} = 11# - {...} = 13# - {...} = 17# - {...} = 29# - {...} = 31# - {...} = 43# - {...} = 89# - {...} = 101# - {...}
151= 7# - {...} = 13# - {...} = 29# - {...} = 37# - {...} = 47# - {...} = 107# - {...} = 149# - {...}
157= 7# - {...} = 11# - {...} = 13# - {...} = 19# - {...}
163= 7# - {...} = 13# - {...} = 37# - {...} = 59# - {...} = 67# - {...}
167= 7# - {...} = 11# - {...} = 13# - {...} = 43# - {...} = 47# - {...} = 59# - {...} = 79# - {...} = 101# - {...}
173= 7# - {...} = 11# - {...} = 29# - {...} = 31# - {...} = 157# - {...}
179= 7# - {...} = 11# - {...} = 13# - {...} = 17# - {...} = 19# - {...} = 29# - {...} = 71# - {...} = 131# - {...}
181= 7# - {...} = 11# - {...} = 19# - {...} = 23# - {...} = 41# - {...} = 73# - {...}
191= 7# - {...} = 17# - {...} = 53# - {...} = 59# - {...} = 73# - {...} = 79# - {...} = 173# - {...}
193= 7# - {...} = 13# - {...} = 29# - {...} = 41# - {...} = 53# - {...}
197= 7# - {...} = 11# - {...} = 13# - {...} = 23# - {...} = 31# - {...} = 41# - {...} = 83# - {...} = 107# - {...} = 151# - {...}
199= 7# - {...} = 11# - {...} = 17# - {...} = 23# - {...} = 67# - {...} = 71# - {...}
211= 11# - {...} = 13# - {...} = 17# - {...} = 31# - {...} = 107# - {...}
223= 11# - {...} = 17# - {...} = 61# - {...} = 127# - {...} = 131# - {...} = 179# - {...} = 181# - {...}
227= 11# - {...} = 13# - {...} = 29# - {...} = 31# - {...} = 41# - {...} = 211# - {...}
229= 11# - {...} = 19# - {...} = 43# - {...} = 53# - {...}
233= 67# - {...} = 149# - {...} = 167# - {...}
239= 17# - {...} = 19# - {...} = 23# - {...} = 37# - {...} = 41# - {...} = 71# - {...} = 79# - {...} = 101# - {...} = 163# - {...}
241= 11# - {...} = 13# - {...} = 19# - {...} = 29# - {...} = 61# - {...} = 71# - {...}
251= 31# - {...} = 41# - {...} = 79# - {...} = 97# - {...} = 113# - {...}
257= 11# - {...} = 17# - {...} = 19# - {...} = 29# - {...} = 31# - {...} = 37# - {...} = 73# - {...} = 97# - {...} = 113# - {...} = 127# - {...} = 167# - {...} = 229# - {...}
263= 17# - {...} = 37# - {...} = 43# - {...} = 61# - {...} = 71# - {...} = 83# - {...} = 107# - {...} = 109# - {...}
269= 13# - {...} = 17# - {...} = 23# - {...} = 31# - {...} = 61# - {...}
271= 11# - {...} = 13# - {...} = 31# - {...} = 59# - {...} = 79# - {...} = 89# - {...}
277= 13# - {...} = 17# - {...} = 29# - {...} = 47# - {...} = 59# - {...} = 89# - {...} = 113# - {...}
281= 11# - {...} = 19# - {...} = 37# - {...} = 43# - {...} = 67# - {...} = 113# - {...} = 139# - {...}
283= 11# - {...} = 17# - {...} = 23# - {...} = 31# - {...} = 37# - {...} = 107# - {...}
293= 11# - {...} = 17# - {...} = 23# - {...} = 31# - {...} = 43# - {...} = 53# - {...} = 103# - {...} = 191# - {...} = 197# - {...}
307= 11# - {...} = 13# - {...} = 17# - {...} = 29# - {...} = 37# - {...} = 61# - {...} = 233# - {...} = 263# - {...}
311= 11# - {...} = 17# - {...} = 23# - {...} = 37# - {...} = 67# - {...} = 97# - {...} = 223# - {...}
313= 11# - {...} = 13# - {...} = 19# - {...} = 29# - {...} = 41# - {...} = 47# - {...} = 61# - {...} = 67# - {...} = 73# - {...} = 109# - {...} = 223# - {...} = 229# - {...} = 23
9# - {...}
317= 11# - {...} = 181# - {...} = 257# - {...} = 277# - {...}
331= 11# - {...} = 17# - {...} = 19# - {...} = 47# - {...} = 53# - {...} = 67# - {...} = 79# - {...} = 311# - {...}
337= 11# - {...} = 29# - {...} = 53# - {...} = 97# - {...}
347= 13# - {...} = 29# - {...} = 43# - {...} = 71# - {...} = 211# - {...} = 251# - {...}
349= 19# - {...} = 37# - {...} = 41# - {...} = 43# - {...} = 223# - {...} = 239# - {...}
353= 17# - {...} = 23# - {...} = 29# - {...} = 31# - {...} = 41# - {...} = 73# - {...} = 83# - {...}
359= 11# - {...} = 13# - {...} = 19# - {...} = 113# - {...}
367= 13# - {...} = 19# - {...} = 29# - {...} = 47# - {...} = 181# - {...} = 263# - {...} = 337# - {...}
373= 17# - {...} = 23# - {...} = 37# - {...} = 61# - {...} = 67# - {...} = 109# - {...} = 127# - {...} = 139# - {...} = 149# - {...} = 227# - {...} = 367# - {...}
379= 11# - {...} = 43# - {...} = 61# - {...} = 109# - {...} = 313# - {...}
383= 17# - {...} = 41# - {...} = 73# - {...} = 79# - {...} = 109# - {...} = 179# - {...} = 239# - {...} = 263# - {...}
389= 13# - {...} = 17# - {...} = 23# - {...} = 47# - {...} = 67# - {...} = 79# - {...} = 103# - {...} = 137# - {...}
397= 11# - {...} = 13# - {...} = 31# - {...} = 53# - {...} = 61# - {...} = 73# - {...} = 109# - {...} = 131# - {...} = 367# - {...}
401= 13# - {...} = 19# - {...} = 23# - {...} = 41# - {...} = 197# - {...} = 337# - {...}
409= 11# - {...} = 17# - {...} = 47# - {...} = 59# - {...} = 163# - {...} = 173# - {...} = 337# - {...}
419= 13# - {...} = 29# - {...} = 41# - {...} = 61# - {...} = 67# - {...} = 131# - {...} = 139# - {...}
421= 11# - {...} = 17# - {...} = 29# - {...} = 73# - {...} = 89# - {...} = 139# - {...}
431= 11# - {...} = 13# - {...} = 17# - {...} = 29# - {...} = 37# - {...} = 43# - {...} = 59# - {...} = 79# - {...} = 109# - {...} = 163# - {...} = 281# - {...} = 337# - {...}
433= 11# - {...} = 17# - {...} = 173# - {...} = 431# - {...}
439= 11# - {...} = 19# - {...} = 23# - {...} = 31# - {...} = 37# - {...} = 113# - {...} = 163# - {...} = 167# - {...} = 229# - {...} = 419# - {...}
443= 11# - {...} = 13# - {...} = 17# - {...} = 29# - {...} = 83# - {...} = 131# - {...} = 163# - {...} = 233# - {...} = 353# - {...} = 367# - {...} = 397# - {...}
449= 11# - {...} = 13# - {...} = 17# - {...} = 31# - {...} = 37# - {...} = 103# - {...} = 137# - {...} = 197# - {...} = 331# - {...} = 349# - {...} = 401# - {...}
457= 13# - {...} = 19# - {...} = 41# - {...} = 47# - {...} = 109# - {...} = 199# - {...} = 349# - {...}
461= 13# - {...} = 17# - {...} = 31# - {...} = 47# - {...} = 59# - {...} = 101# - {...} = 311# - {...} = 313# - {...}
463= 11# - {...} = 13# - {...} = 17# - {...} = 47# - {...} = 61# - {...} = 101# - {...} = 151# - {...} = 199# - {...} = 281# - {...} = 367# - {...} = 419# - {...}
467= 29# - {...} = 37# - {...} = 43# - {...} = 109# - {...} = 271# - {...} = 373# - {...}
479= 11# - {...} = 17# - {...} = 19# - {...} = 37# - {...} = 53# - {...} = 59# - {...} = 157# - {...} = 233# - {...} = 283# - {...}
487= 11# - {...} = 19# - {...} = 71# - {...} = 163# - {...}
491= 19# - {...} = 41# - {...} = 101# - {...} = 113# - {...} = 149# - {...} = 157# - {...} = 239# - {...}
499= 11# - {...} = 13# - {...} = 19# - {...} = 103# - {...} = 139# - {...} = 157# - {...} = 163# - {...} = 359# - {...}
用时 0.25401 秒
其中7、11、13、17为满解，即可行解全部是有效解；且只有7、11、31少于3个解。

aimisiyou 发表于 2024-3-23 17:37:49

xbtianlang 发表于 2024-3-23 13:52
2024-03-23 13:49:22
7= 5# - {...}
11= 5# - {...} = 7# - {...}


一切皆有可能！

northwolves 发表于 2024-3-24 12:48:21

p<10000的唯2解：
11        2        {5,7}

northwolves 发表于 2024-3-24 12:48:44

p<10000的3个解组合：

13        3        {5,7,11}
19        3        {5,7,13}
23        3        {5,11,19}
29        3        {7,11,17}
37        3        {7,11,19}
61        3        {7,17,23}
233        3        {67,149,167}
3191        3        {13,211,2069}
8161        3        {61,373,3607}

northwolves 发表于 2024-3-24 12:49:23

p<10000的4个解组合：

17        4        {5,7,11,13}
41        4        {11,13,19,29}
43        4        {7,11,19,23}
53        4        {7,17,19,41}
67        4        {11,19,29,47}
71        4        {7,11,13,41}
79        4        {7,19,23,31}
83        4        {7,13,31,79}
109        4        {7,13,17,59}
157        4        {7,11,13,19}
229        4        {11,19,43,53}
317        4        {11,181,257,277}
337        4        {11,29,53,97}
359        4        {11,13,19,113}
433        4        {11,17,173,431}
487        4        {11,19,71,163}
1091        4        {37,83,137,619}
2113        4        {11,13,89,1321}
2129        4        {11,13,37,1439}
2677        4        {23,53,137,617}
3769        4        {13,19,97,383}
4099        4        {13,67,439,661}
7001        4        {13,769,1741,3271}
8353        4        {19,43,59,101}

northwolves 发表于 2024-3-24 13:58:08

p=36263 ,5个解：{3853, 5197, 5209, 6247, 6397}

northwolves 发表于 2024-3-24 13:59:29

p<10000 唯一解：
7      1      {5}
31      1      {7}

xbtianlang 发表于 2024-3-26 10:00:57

northwolves 发表于 2024-3-24 13:58
p=36263 ,5个解：{3853, 5197, 5209, 6247, 6397}

2024-03-24 16:37:54
... ...
-3846-36241
-3847-36251
36263 {3853, 5197, 5209, 6247, 6397, 6827, 11681}
用时 23836.19552 秒
检验到36251#，用时很长。

northwolves 发表于 2024-4-20 22:12:53

本帖最后由 northwolves 于 2024-4-20 22:17 编辑

假设$a_n$为大于1的最小正整数 $k$，使得恰好存在$n$个形式为$m!−k$的素数，如何快速计算出$a_{100}$?

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