大猜想，估计无人能够破解的猜想

Treenewbee · 发表于 2024-4-27 16:48

不错的题目，不过找到了5个反例：{18, 63, 71, 165, 265}

Treenewbee · 发表于 2024-4-27 16:57

f[n_]:=Select[Prime@Range[PrimePi@(n^2)+1,PrimePi@((n+1)^2+1)],PrimeQ[n^2+(n+2)^2-#]&];Table[{n,f[n]},{n,300}]//MatrixForm

复制代码

n P1
--------------------------------------
1 {3,5}
2 {7}
3 {11,17}
4 {23}
5 {31,37}
6 {41,47}
7 {59}
8 {67}
9 {89,101}
10 {107,113}
11 {127,139}
12 {149,167}
13 {197}
14 {211,223}
15 {233,251,257}
16 {263,269}
17 {313}
18 {}
19 {383,401}
20 {421}
21 {449,461,467,479}
22 {491,503}
23 {541,547,577}
24 {593,599}
25 {653,677}
26 {691,709,727}
27 {743,761,773}
28 {797,821,827}
29 {883}
30 {911,941,947,953}
31 {1019}
32 {1051,1063,1087}
33 {1091,1097,1151}
34 {1163,1193,1223}
35 {1291,1297}
36 {1301,1307,1367}
37 {1409,1439}
38 {1447}
39 {1583,1601}
40 {1667}
41 {1699,1741,1747,1753}
42 {1787,1811,1823}
43 {1871,1877,1901}
44 {1999}
45 {2027,2081}
46 {2153,2207}
47 {2221,2239,2269}
48 {2357,2381,2393}
49 {2411,2423,2459}
50 {2521,2557}
51 {2609,2621,2633,2657,2699}
52 {2711,2741,2777,2801}
53 {2833,2917}
54 {2963,2969,3011}
55 {3083,3137}
56 {3169,3181,3187,3229}
57 {3323,3359}
58 {3371,3407}
59 {3511,3529,3559,3571}
60 {3623,3677}
61 {3761,3767,3779}
62 {3847,3889,3919}
63 {}
64 {4211}
65 {4231,4273,4357}
66 {4397,4457,4463}
67 {4517,4547}
68 {4663,4723}
69 {4793,4799,4871}
70 {4931,4937,5003}
71 {}
72 {5189,5261,5273,5279,5309}
73 {5381,5471,5477}
74 {5503,5563,5569}
75 {5651,5657,5693,5711,5741}
76 {5807,5813,5849,5879}
77 {5953,6007,6037,6079}
78 {6131,6173,6197,6221}
79 {6311,6329,6353}
80 {6421,6451,6547,6553}
81 {6581,6659,6689}
82 {6737,6761,6779,6803,6833,6863,6869}
83 {6907,7057}
84 {7103,7121}
85 {7247,7253,7307}
86 {7417,7459,7537,7549}
87 {7583,7589,7607,7649,7673}
88 {7757,7907}
89 {7933,7993,8011,8101}
90 {8117,8273}
91 {8387,8429}
92 {8521,8539,8563,8581,8623}
93 {8663,8741,8807,8837}
94 {8849,9011}
95 {9043,9091,9151,9157}
96 {9281,9323,9341}
97 {9419,9461,9467,9491,9521,9533,9587}
98 {9631,9697,9721,9733,9787}
99 {9833,9839,9851,9923,9941}
100 {10061,10067,10091,10103,10133,10151,10181,10193}
101 {10243,10333,10357}
102 {10487,10529,10589,10607}
103 {10631,10781}
104 {10891,10903,10939,10993}
105 {11027,11213}
106 {11243,11279,11321,11351,11411}
107 {11467,11491,11497,11503,11551,11587}
108 {11777,11783,11831,11867}
109 {11933,11939,12041,12101}
110 {12157,12211,12253,12301}
111 {12347,12377,12401,12437,12479}
112 {12641,12647}
113 {12823}
114 {13001,13121}
115 {13337,13451,13457}
116 {13477,13591,13669,13687}
117 {13691,13697,13763,13799,13841,13883}
118 {13997,14081}
119 {14173,14251,14323,14341,14401}
120 {14537,14543,14561,14627}
121 {14669,14717,14753,14813,14831,14879}
122 {14887,14929,14947,14983,15061,15073,15121}
123 {15173,15227,15287,15377}
124 {15443,15461,15569,15581}
125 {15643,15667,15817,15877}
126 {15959,16007,16067,16073}
127 {16139,16217,16223,16319,16349}
128 {16453,16633}
129 {16703,16823,16871,16901}
130 {16931,16937,17033,17093,17117}
131 {17167,17191,17299,17341,17359,17383,17401,17419}
132 {17471,17477,17489,17573,17597}
133 {17783,17837,17957}
134 {17959,17971,18013,18199,18223}
135 {18251,18401}
136 {18503,18539,18593,18671,18701,18743}
137 {18859,19009}
138 {19073,19181,19211}
139 {19463,19541}
140 {19603,19717,19753,19801}
141 {19889,19919,19937,19961,19973,19997,20129,20147}
142 {20183,20219,20261,20357,20393}
143 {20731}
144 {20873,20903,20963}
145 {21143,21227,21317}
146 {21379,21433,21481,21493,21559}
147 {21617,21701,21737,21773,21881}
148 {22013,22037,22133,22157}
149 {22303,22381,22453,22501}
150 {22541,22643,22697,22727,22751,22787}
151 {22811,22853,22871,22877,23021}
152 {23131,23143,23197,23227,23311}
153 {23561,23603,23633,23687}
154 {23801,23813,23873,23981,24023}
155 {24043,24103,24337}
156 {24533,24551,24623}
157 {24677,24683,24767,24809,24917,24953}
158 {25111,25117,25153,25243,25261}
159 {25409,25439,25469,25523,25601}
160 {25667,25673,25703,25733,25841,25847,25913}
161 {25933,25951,26041,26053,26083,26119,26227}
162 {26249,26261,26339,26357,26417,26423,26459,26513}
163 {26597,26687,26717,26777,26783,26801,26813,26891}
164 {27043,27091,27211}
165 {}
166 {27617,27749,27779,27827}
167 {27901,28057,28099}
168 {28307,28463,28493,28517}
169 {28571,28649,28793,28901}
170 {29101,29137,29173}
171 {29243,29297,29303,29333,29411,29429,29453,29501,29537}
172 {29663,29723,29741,29789}
173 {30187,30241}
174 {30341,30449,30539}
175 {30677,30707,30773,30803,30941,30971,30977}
176 {31033,31147,31183,31267,31327}
177 {31379,31397,31511,31601,31643,31649}
178 {31721,31847,32027}
179 {32083,32089,32191,32233,32299,32323,32359,32401}
180 {32411,32441,32531,32537,32693}
181 {32771,32789,32933,32939,33071}
182 {33151,33211,33223,33301,33343,33391,33403,33487}
183 {33503,33587,33773,33791,33851,33857}
184 {33863,33941,34031}
185 {34231,34297,34351,34501,34543,34591}
186 {34613,34649,34673,34739,34781,34841,34871,34913}
187 {35099,35117,35153,35159,35267,35327}
188 {35407,35437,35461,35521,35533,35593}
189 {35729,35933,35951,35993,36011}
190 {36107,36131,36251,36293,36467}
191 {36529,36541,36559,36571,36607,36643,36691,36709,36787}
192 {36929,37139,37223}
193 {37277,37307,37463,37493}
194 {37813}
195 {38183,38231,38273,38333}
196 {38459,38501,38699,38729,38747}
197 {38959,38971,39043,39097,39181}
198 {39233,39251,39317,39341,39383,39443,39581}
199 {39659,39719,39749,39761,39839,39971,39989}
200 {40111,40177,40213}
201 {40427,40433,40493,40529,40559,40637,40763,40787}
202 {40823,40841,41039,41177,41189}
203 {41221,41281,41341,41593,41617}
204 {41729,41759,41771,41813,41843,41969,41981}
205 {42101,42131,42197,42407,42437}
206 {42463,42499,42649,42697,42841}
207 {42923,42953,42989,43013,43049,43103,43133}
208 {43391,43403,43451,43577}
209 {43711,43753,43933,44071,44101}
210 {44201,44267,44273,44357,44501,44507}
211 {44549,44657,44699,44711,44729,44753,44771,44927}
212 {44983,45127,45259,45307,45337}
213 {45413,45491,45503,45533,45641,45707,45767}
214 {45833,45863,45893,45953,46103,46181}
215 {46273,46381,46447,46507,46567,46591,46633}
216 {46679,46817,46829,46877,46901,47051,47057,47087}
217 {47111,47147,47207,47309,47339,47351,47441,47459,47507}
218 {47527,47653,47737,47947}
219 {47981,48023,48179,48239,48311,48353}
220 {48407,48473,48491,48527,48563,48647,48731}
221 {48859,48889,48907,48973,49033,49177,49201,49207}
222 {49307,49331,49367,49391,49409,49523}
223 {49811,49937,49943,49991,50021,50033,50093,50123,50147,50177}
224 {50221,50263,50329,50359,50581}
225 {50651,50723,50741,50867,50891,50951,50957}
226 {51131,51257,51263,51341,51347,51413,51461,51479,51521}
227 {51607,51679,51721,51769,51787,51817,51913,51949}
228 {52067,52127,52163,52253,52301,52313}
229 {52571,52631,52673,52709,52733,52883,52901}
230 {52951,53101,53113,53173,53197,53323}
231 {53381,53549,53591,53639,53657,53699,53759,53819}
232 {53951,54059,54083,54167,54269,54287}
233 {54367,54493,54541,54727}
234 {54779,54833,54941,54983,55079,55109}
235 {55313,55631,55661,55673,55697}
236 {55711,55807,55813,55837,55897,55903,56041,56101,56131}
237 {56171,56249,56333,56369,56393,56417,56477,56543}
238 {56687,56897,56957,56993,57041}
239 {57259,57301,57349,57373,57493,57601}
240 {57653,57713,57737,57773,57947,58013}
241 {58109,58151,58193,58217,58229,58367,58403,58451}
242 {58603,58657,58693,58741,59023}
243 {59093,59123,59351,59447,59453,59513}
244 {59669,59699,59951}
245 {60091,60133,60223,60271,60331,60373,60397,60427}
246 {60527,60611,60617,60689,60737,60869,60899}
247 {61031,61043,61253,61343,61379,61463}
248 {61507,61603,61657,61681,61813,61861,61933,61987}
249 {62081,62099,62129,62141,62201,62501}
250 {62627,62687,62723,62873}
251 {63097,63103,63211,63313,63361,63409,63421}
252 {63521,63587,63617,63647,63719,63737,63803,63863,63929,64007}
253 {64037,64157,64217,64223,64271,64373,64433}
254 {64633,64783,64849,64879,64951}
255 {65027,65033,65111,65123,65147,65357,65423,65537}
256 {65609,65633,65651,65687,65717,65921,65927,65963,65993,66029}
257 {66109,66529}
258 {66587,66617,66653,66683,66821,66947,66977,67043,67061}
259 {67103,67271,67349,67493,67523,67601}
260 {67723,67801,67807,67933}
261 {68141,68147,68171,68261,68279,68399,68477}
262 {68687,68909,69119,69143,69149}
263 {69193,69403,69463,69697}
264 {69833,70001,70079,70139,70163,70181,70223}
265 {}
266 {70867,71011,71143,71161,71167,71191,71233,71263,71287}
267 {71363,71399,71429,71483,71549,71597,71663,71741,71789,71807}
268 {72053,72077,72101,72173,72221,72227,72341}
269 {72493,72559,72613,72661,72739,72763,72871,72901}
270 {72911,72923,72977,73037,73061,73127,73133,73277,73331,73361}
271 {73517,73529,73589,73607,73613,73673,73751,73877,73943}
272 {74047,74101,74131,74203,74353,74449,74509}
273 {74597,74717,74747,74831}
274 {75149,75161,75269,75431,75479,75521,75611}
275 {75703,75793,75883,75913,75931,75967,76123,76147}
276 {76259,76289,76367,76379,76511,76541,76631,76679}
277 {76757,76847,76871,76883,76949,77081,77093,77153,77201,77279}
278 {77317,77377,77383,77491,77521,77527,77563}
279 {77999,78179,78233,78401}
280 {78497,78623,78737,78791,78893}
281 {78979,79063,79111,79147,79273,79357,79393,79423}
282 {79559,79613,79691,79811,79817,79901,79907,79973,80039}
283 {80111,80141,80231,80273,80537,80567,80627,80657}
284 {80683,80749,80803,80833,80989,81013,81031,81043}
285 {81233,81293,81353,81371,81401,81527,81563}
286 {81929,81953,82013,82139,82241}
287 {82393,82483,82507,82549,82591,82633,82657,82813}
288 {82997,83231,83267,83423}
289 {83813,83903,83939,84011,84059}
290 {84121,84127,84163,84673}
291 {84701,84713,84737,84827,84869,85061,85091,85103,85199}
292 {85331,85451,85517,85847}
293 {86017,86131,86197,86287,86341,86413}
294 {86561,86579,86693,86729,86753,86771,86969,86981,87011}
295 {87257,87317,87323,87491}
296 {87631,87649,87679,87691,87739,87811,88093}
297 {88211,88223,88337,88379,88397,88523,88589,88607,88793,88799}
298 {88937,89021,89051,89123,89213,89237,89303,89387}
299 {89491,89521,89533,89563,89599,89689,89839,89983,90001}
300 {90011,90053,90107,90227,90371,90401,90473,90527}

lusishun · 发表于 2024-4-27 18:11

lusishun 发表于 2018-1-14 08:44
》》》》两个相邻的奇（偶）数的平方之和，定是这两平方数之间的两素数之和。
注意是相邻的，
1与3是相 ...

大T先生，发表了五个反例推翻了这猜想。告一段。

大T先生进步提出，估计就这五个反例。
这又是一个很好的猜想。

Treenewbee · 发表于 2024-4-27 21:48

本帖最后由 Treenewbee 于 2024-4-27 21:50 编辑

lusishun
以18为例，扩大范围，在17^2=289，至21^2=441之间一定存在两素数，它们的之和等于偶数324+400=724。发表于 2024-4-27 21:15
---------------------------------
扩大到n-1--->n+3这5个反例就都不存在了

白新岭 · 发表于 2024-4-27 22:37

范围统计本段内
5000 0 0
5100 1 1
5200 1 0
5300 3 2
5400 5 2
5500 8 3
5600 11 3
5700 12 1
5800 16 4
5900 19 3
6000 22 3
6100 25 3
6200 26 1
6300 29 3
6400 31 2
6500 32 1
6600 33 1
6700 34 1
6800 37 3
6900 39 2
7000 43 4
7100 45 2
7200 47 2
7300 49 2
7400 50 1
7500 52 2
7600 58 6
7700 62 4
7800 66 4
7900 66 0
8000 69 3
8100 72 3
8200 74 2
8300 75 1
8400 77 2
8500 79 2
8600 82 3
8700 86 4
8800 88 2
8900 92 4
9000 94 2
9100 96 2
9200 99 3
9300 104 5
9400 106 2
9500 112 6
9600 116 4
9700 117 1
9800 121 4
9900 125 4
10000 127 2
这是用1万这个偶数做的分析，分析结果，不在意料之中，单计数，1万共有127对素数对，分成50个区间段，大概每段有2.54对，即2对，4对，3对都正确（毕竟素数对没有半对）

白新岭 · 发表于 2024-4-27 22:45

素数对区间数
0 2
1 9
2 16
3 11
4 9
5 1
6 2
落到2,3,4对上的区间段占36组，其余占14，没有素数对的2个区间段，有5对的有1个区间段，有6对的有2个区间段，只有1对的是例外，有9个区间段，基本上符合正态分布，越边缘，越少。
所以，lusishun的猜想，在n大于一定值后成立。（当然首先要证明哥德巴赫猜想）。

白新岭 · 发表于 2024-4-27 22:48

Treenewbee 发表于 2024-4-27 21:48
lusishun
以18为例，扩大范围，在17^2=289，至21^2=441之间一定存在两素数，它们的之和等于偶数324+400= ...

我希望Treenewbee先生，能把100万这个偶数，像我那样分析出各区间段上的数量，素数对数量接近平均值的组数，分布情况给出，使人一目了然。

白新岭 · 发表于 2024-4-27 23:01

两个相邻的奇（偶）数的平方之和，定是这两平方数之间的两素数之和
如果上述猜想成立，则在每个区间段上都成立，即
\([(2n)^2,(2n+2)^2],或[(2n-1)^2,(2n+1)^2]\),划分的（n+1）个区间段上，都至少有一对素数对，是其和等于它们的和值。也可以大概估计出它们的素数对，即把偶数本身分成大概根号N份。例如10000，分成100份，则占它的1/100.

lusishun · 发表于 2024-4-28 07:01

整理一下：
（鲁思顺）大猜想：
在区间[n^2,(n+2)^2]内，存在两个素数，其和等于n^2+(n+2)^2.

大T先生猜想：（鲁思顺）大猜想中有且只有五个反例。

只为方便，便于区别，叙述，设暂用名称，相信大家会理解。

lusishun · 发表于 2024-4-28 14:11

lusishun 发表于 2024-4-27 23:01
整理一下：
（鲁思顺）大猜想：
在区间[n^2,(n+2)^2]内，存在两个素数，其和等于n^2+(n+2)^2.

大猜想的有关内容在这里。
看上贴

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