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[讨论] 素阶乘猜想：p = P# - q

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xbtianlang
发表于 2024-3-22 23:04:51 | 显示全部楼层 |阅读模式

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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

毋因群疑而阻独见　　毋任己意而废人言
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xbtianlang
 楼主| 发表于 2024-3-23 13:52:17 | 显示全部楼层
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。

点评

xbtianlang
限于篇幅，具体的素数显示已省略。  发表于 2024-3-23 13:53
毋因群疑而阻独见　　毋任己意而废人言
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aimisiyou
发表于 2024-3-23 17:37:49 | 显示全部楼层
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！
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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northwolves
发表于 2024-3-24 12:48:21 | 显示全部楼层
p<10000的唯2解：
11        2        {5,7}
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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northwolves
发表于 2024-3-24 12:48:44 | 显示全部楼层
p<10000的3个解组合：

  1. 13        3        {5,7,11}
  2. 19        3        {5,7,13}
  3. 23        3        {5,11,19}
  4. 29        3        {7,11,17}
  5. 37        3        {7,11,19}
  6. 61        3        {7,17,23}
  7. 233        3        {67,149,167}
  8. 3191        3        {13,211,2069}
  9. 8161        3        {61,373,3607}
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毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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northwolves
发表于 2024-3-24 12:49:23 | 显示全部楼层
p<10000的4个解组合：

  1. 17        4        {5,7,11,13}
  2. 41        4        {11,13,19,29}
  3. 43        4        {7,11,19,23}
  4. 53        4        {7,17,19,41}
  5. 67        4        {11,19,29,47}
  6. 71        4        {7,11,13,41}
  7. 79        4        {7,19,23,31}
  8. 83        4        {7,13,31,79}
  9. 109        4        {7,13,17,59}
  10. 157        4        {7,11,13,19}
  11. 229        4        {11,19,43,53}
  12. 317        4        {11,181,257,277}
  13. 337        4        {11,29,53,97}
  14. 359        4        {11,13,19,113}
  15. 433        4        {11,17,173,431}
  16. 487        4        {11,19,71,163}
  17. 1091        4        {37,83,137,619}
  18. 2113        4        {11,13,89,1321}
  19. 2129        4        {11,13,37,1439}
  20. 2677        4        {23,53,137,617}
  21. 3769        4        {13,19,97,383}
  22. 4099        4        {13,67,439,661}
  23. 7001        4        {13,769,1741,3271}
  24. 8353        4        {19,43,59,101}
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毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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northwolves
发表于 2024-3-24 13:58:08 | 显示全部楼层
p=36263 ,5个解：{3853, 5197, 5209, 6247, 6397}
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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northwolves
发表于 2024-3-24 13:59:29 | 显示全部楼层
p<10000 唯一解：
7        1        {5}
31        1        {7}
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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xbtianlang
 楼主| 发表于 2024-3-26 10:00:57 | 显示全部楼层
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。

点评

northwolves
辛苦！除了硬算，似乎也没有好办法  发表于 2024-3-26 10:48
毋因群疑而阻独见　　毋任己意而废人言
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northwolves
发表于 2024-4-20 22:12:53 | 显示全部楼层
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$a_{100}$?
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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