关于 Coldbach问题的证明

倪家泰

(攀枝花市第十五中学,四川 攀枝花 617063)

关于 Coldbach问题的证明

倪家泰

(攀枝花市第十五中学,四川 攀枝花 617063)

Coldbach问题是著名的世界数学难题,一百多年以来未曾有人能证明它。用初等数论能不能证明这一问题,现在也争论不休。我的论文成功的证明了这一问题,并且解决了这个争论。

本文以自然数 N>2为等差中项,寻求素公差 d。对于 3≤N≤421的等差中项,一一列举其素公差,对于N>421的等差中项,证明其素公差存在。孪生素数对问题是哥氏问题的特殊情况,同时进行证明。

Coldbach问题;素公差;同余式;孙子定理;孪生素数

定义 1 自然数N>2,若存在非负数 d为公差,使N-d和N+d为二素数,则称 d为N之素公差。

定义2 称同余式

n≡r(r与 N和 -N不同余)modq(0≤r

N关于 2的补同余式有且仅有一个。若 q|N,N关于 q的补同余式有 q-1个:若 q不能整除N,N关于 q的补同余式有 q-2个。

当 d〈N-qk时,d和 ±N对所有小于全体素数都不同余,故得到下列同余式方程组:

称此方程组为N的补同余式组。

由孙子定理,N的每一补同余式组在 0到 q1q2……qk间有且仅有一解 dN。

dN为 N的补同余式组 (1)之解,dN就满足方程中的每一同余式,dN满足第 i(i=1, 2……k)个同余式,qi不能整除N-dN,且 qi不能整除N+dN,即 qi既不是N-dN的因数,也不是N+dN的因数,换言之素数列 (Ⅰ)中的任何一个素数既不能整除 N-dN,也不能整除N+dN,因而N-dN和N+dN都是素数。

是N的补同余式组 (1)之从小到大排列的解。

定义 3 令 b1=c1,bk-bk-1=ck称 ck为 N的补同余式组 (1)之解差。解差中的最大者称N补同余式组之最大解差,简称N的最大解差。

例 1 求N=62时同余式组 (1)之全体解及其最大解差和平均解差?

∴素数列 (Ⅰ) =2,3,5,7,11

62关于 2,3,5,7,11的余数分别 0,2,2,6,7。62的补同余式组 (1)为

n≡r1(r1=1) mod 2

n≡r2(r2=0) mod 3

n≡r3(r3=0,1,4) mod 5

n≡r4(r4=0,2,3,4,5) mod 7

n≡r5(r5=0,1,2,3,5,6,8,9,10) mod 11

其解从小到大排列如下:

9,21,39,45,75,105,129,135,159,165,171,189,201,219,231,255, 261,285,291,339,369,375,399,429,441,459,465,471,495,501,525,549, 555,585,591,621,639,651,669,681,705,735,759,765,789,795,801,819, 831,849,861,879,885,891,915,921,945,969,999,1011,1029,1059,1089, 1095,1101,1125,1131,1155,1179,1185,1209,1215,1221,1251,1281,1299, 1311,1341,1365,1389,1395,1419,1425,1431,1449,1461,1479,1491,1509, 1515,1521,1545,1551,1575,1605,1629,1641,1659,1671,1689,1719,1725, 1755,1761,1785,1809,1815,1839,1845,1851,1869,1881,1911,1935,1941, 1971,2019,2025,2049,2055,2079,2091,2109,2121,2139,2145,2151,2175, 2181,2205,2235,2265,2271,2289,2301。

其最大解差为 48。

引理 2 当N>420时,数N的补同余式组 (1)之最大解差小

证明:数N的补同余式组之解必须满足前二个同余式

n≡r1(r1与N不同余) mod q1

n≡r2(r2与N和 -N不同余) mod q2

若这两个同余式构成的同余式组之解为 a(当 3不能整除N时,r2=0,a的值有且仅有一个;当 3|N,r2可为 1或 2,a的值有二个),则 N的补同余式组之解具有 a+6m的形式。

对于 a的一个固定值,N的补同余式组解中,具有 a+6m形式之解的最大解差中不是解的数之个数,是m从某自然数起连续几个自然数的个数。这些数必然是满足下同余式中至少一个同余式。

由于 (2)中同余式个数最多有 2(k-2)个,不重复满足 (2)中同余式之一的情况下,这些第一次满足 (2)中同余式之数在从起不是解的数中,不妨记这些数从小到大排列为

为保证 (2)中同余式的第一次满足最多,我们约定 (3)中若(1≤i≤l)同时为qt和 qr(3≤t

在此基础上研究重复满足 (2)中同余式之一的情况个数。重复满足 (2)中同余式之一的数只需插入其中间,插入情况及个数为:若(1≤j≤l)满足 (2)中同余式 n≡rimod qi,当 qi|N时,第一个重复满足数前有 qi个 (2)中同余式满足数。其余每隔 qi-1个 (2)中同余式满足数有 1个重复满足数:当 qi不能整除N时,第一个重复满足数前有 qi个 (2)中同余式满足数,其余每隔 qi-2个 (2)中同余式满足数有 2个重复满足数。显然 qi|N时的重复满足数个数没有 qi不能整除N时多,所以计算最大解差中重复满足数个数时均以 qi不能整除N的情况来估计 (2)中所有同余式的重复满足个数。

为了使最大解差估计中重复满足数最多,我们约定,每次计算时若末尾的个数在 1到qi-3内,不论多少均估计有一个重复满足数。且依下方法步骤计算:假设满足 (2)中第一个同余式,重复满足数个数计算得 l1,插入这 l1个数后排列 (3)有 l+l1个新排列数,再设满足 (2)中第二个同余式,因和第二次的重复满足数之计算无关,故以1个排列数来计算其重复满足数个数,……像这样依次计算 (2)中各同余式之重复满足数个数,并在此基础上适当调整顺序以使重复满足数个数达到最多。

例 2 求 k=30时,(2)中各同余式重复满足数个数和相应的最大解差估计值?

解 k=30,qk=113,2(k-2) =56, (2)中各同余式重复满足数计算情况及个数如下:

计算式 个数 计算式 个数56=5+317 35 322=47+456+5 14 90=7+516+3 34 335=109+1072+12 6 123=11+912+4 26 340=113+1112+5 6 148=13+1112+3 26 345=67+654+18 10 173=17+1510+6 22 354=71+694+7 10 194=19+1710+5 22 363=53+516+4 14 215=23+219+3 20 376=89+873+26 8 234=31+297 15 383=97+953+1 8 248=29+278+3 18 390=79+774+3 10 265=37+356+18 14 399=101+993+1 8 278=41+396+3 14 406=73+714+49 10 291=43+416+2 14 415=83+814+8 10 304=59+574+17 10 424=107+1053+2 8 313=61+594+16 10 431=103+1013+27 8

(2)中各同余式满足数为 431+8+27=466

记N补同余式之解的最大解差中满足 (2)中同余式的数之个数为 f(k),则最大解差为 6[f(k) +1]。按上述方法计算不难得到下表:

k 10 11 12 13 14 15 16 17 18 19 20 21 qk29 31 37 41 43 47 53 59 61 67 71 73 6[f(k) +1] 354 450 510 606 720 804 930 1032 1146 1242 1404 1542 1 2(q2k-3) 419 479 683 839 923 1103 1403 1739 1859 2243 2519 2663

22 23 24 25 26 27 28 29 30 31 32 33 34 79 83 89 97 101 103 107 109 113 127 131 137 139 1662 1800 1920 2064 2208 2304 2502 2634 2802 2952 3114 3270 3438 3119 3443 3809 4703 5099 5303 5723 5939 6383 8063 8579 9383 9659

上表中说明引理对 10≤k≤34时相应的N成立的,并且每当 k增加 1时,最大解差中满足 (2)中同余式之数的个数增加数不大于 k+12。如 k从 32增加到 33时最大解差中满足 (2)中同余式之数的个数增加了

事实上,当 k每增加 1,排列 (3)中的初次满足数最多增加 2个,这 2个数最多使(2)中关于 q3到 q9的相应各同余式平均增加 2个重复满足数,合起来共同增加 16个满足(2)中的数。这 16个数又最多使 (2)中关于 q10到 q19的相应各同余式平均增加一个重复满足数,这时最大解差中满足 (2)中的数一共最多增加 26个数,故 (2)中不增加重复满足数的同余式越到后面越多,其增加的重复满足数个数平均不到 1个。所以 (2)中前 k-2个同余式在最大解差中最多增加 k+6个满足 (2)中同余式之数。

k增加 1时,(2)中同余式个数也增加一个20),所以 (2)中后面 qi(i≥20)的相应同余式将有不增加重复满足数情况出现,由于即 k+1从 32起至少连续增加 3个值,上同余式的重复满足数才可能增加2个数,但这时 (2)中不增加重复满足数的同余式也至少增加 2个。当 k+1越大, (2)中不增加重复满足数的同余式就越多。

因此,当 k增加 1时,最大解差中最多增加 (k+6) +6=k+12个满足 (2)中同余式之数。即 f(k+1) -f(k)≤k+12当 k增加 1时,素公差的取值范围也从增大到其范围扩大了

这个同余式的重复满足数个数从 k+1为 21起为 6。

由于 (2)中前 k-2个同余式在最大解差中增加最多 k+6个满足 (2)中同余式之数, k+6和 qk+1-2比较,当 k越大,其比值越小。如 k=31,则 k+6=37,而 qk+1-2=131-

要证 2qk+2≥6(k+12),只需证 qk≥3k+35,不难验证:上式从 k=32,qk=131起成立。所以引理对N>420的全体自然数成立。

若N的补同余式组中的解之 a取二个值,则其解为两组形式如 a+bm的数,其一组解之最大解差小另一组解的数,无论在或不在这个最大解差之中,都不扩大最大解差,故引理成立。

定理 1 大于等于 6的偶数可表示为两奇素数之和。

证明:只需证大于等于 3的自然数都有素公差。

后面附表已列出 2

当N>420时,由引理 2,N的同余式组之解的最大解差小于设 N的补同余式组之解首先出现的最大解差由解 d和 d′构成,d

若 d

若 d

令Ci表示 (2)中取值个数。由于 N的补同余式组之解不大于 q1q1q3……qk,所以全体解差和小于q1q2q2……qk,其平均解差小于N的素公差个数大约有

定理2 孪生素数对有无穷对。

证明:自然数N≥6时,若N有素公差 1,显然N-1和N+1是一对孪生素数,这时N是下列同余式组

由于 k是大于等于 2的任意自然数,故孪生素数对有无穷对。

附 表

表2

N N的素公差126 115,113,107,103,97,85,67,65,55,53,47,37,25,23,13 127 124,114,96,84,66,54,30,24,0 128 123,111,105,99,69,45,39,21 129 122,112,110,100,98,82,70,68,62,50,28,22,20,2 130 127,111,99,93,69,63,51,33,27,21 131 126,120,108,102,60,48,42,18,0 132 125,119,109,101,95,91,79,65,61,59,49,35,31,25,19,5 133 130,96,90,66,30,24,6 134 129,123,117,105,93,63,45,35,3 135 128,122,116,106,104,98,94,92,88,76,64,62,56,46,38,32,28,22,4 136 133,105,93,75,63,57,27 137 134,132,126,120,114,96,90,54,36,30,0 138 133,131,125,119,101,95,91,85,59,55,41,35,29,25,11,1 139 132,102,72,60,42,12,0 140 137,129,123,117,111,99,93,87,57,51,39,33,27,9 141 136,130,128,122,110,100,98,88,82,70,58,52,40,38,32,10 142 139,135,129,99,81,69,39,15 143 138,126,120,114,96,90,84,54,36,30,6 144 137,133,127,125,113,107,97,85,83,55,47,37,35,13,7,5 145 132,126,84,78,66,48,36,18,6 146 135,123,117,105,93,87,45,33 147 134,130,124,116,110,104,94,86,80,76,64,50,48,44,34,20,16,10 148 145,129,81,75,51,45,9 149 144,132,120,108,102,90,78,48,42,18,0 150 143,131,127,121,119,113,107,91,89,83,79,77,61,49,47,43,41,23,13,1 151 120,90,78,72,48,42,12,0 152 141,129,111,105,99,81,45,39,21,15 153 140,124,116,110,86,80,74,70,46,44,40,26,14,4 154 123,117,87,75,57,45,27,3 155 138,126,114,108,102,96,84,72,42,24,18 156 151,137,125,115,113,95,85,83,77,73,67,55,43,25,17,7 157 154,150,120,114,84,54,6,0 158 155,153,135,111,105,99,75,69,21,9 159 154,152,148,122,118,112,98,92,80,70,52,32,22,20,8 160 157,153,147,117,81,63,51,33,21,3 161 156,150,132,120,108,102,90,78,72,30,12 162 155,151,149,145,131,119,115,109,101,95,89,79,65,61,49,35,31,11,5 163 150,144,66,60,36,0 164 153,147,117,105,93,75,63,33,27,15 165 152,148,146,142,128,112,106,104,98,92,86,76,68,64,62,58,34,28,26,16, 14,8,2 166 147,105,63,57,27,15 167 164,150,144,126,114,96,84,66,60,30,0 168 163,149,145,139,125,109,101,95,89,71,65,61,59,55,31,29,11,5 169 162,138,108,102,72,60,42,30,12 170 167,147,141,123,111,99,87,81,69,63,57,21,3 171 166,160,142,140,110,100,98,92,70,68,62,58,40,22,20,8 172 165,159,141,135,105,99,69,21,9 173 144,120,90,84,66,60,24,6,0 174 163,157,143,137,133,107,103,95,77,67,65,37,25,23,17,7

N N的素公差175 172,162,156,138,132,102,96,66,48,36,24,18 176 173,171,135,117,105,93,87,75,63,3 177 172,170,160,154,140,136,134,130,116,104,94,80,74,64,50,46,20,14,4 178 175,171,159,135,99,51,21,5 179 174,168,138,132,90,78,72,48,12,0 180 173,169,167,157,151,137,133,127,113,101,97,91,83,77,71,53,49,31,17, 13,1 181 178,168,150,42,30,18,0 182 177,171,165,135,129,111,99,81,75,69,51,45,15,9 183 176,170,166,164,154,130,124,110,94,86,80,74,56,46,44,16,8 184 165,153,147,123,87,57,45,27 185 182,174,168,162,132,126,96,84,78,72,54,48,12,6 186 173,167,163,145,127,125,107,85,83,77,55,47,37,13,7,5 187 180,150,144,126,120,90,84,36,24,6 188 185,171,165,159,129,105,81,75,51,39,9 189 184,178,170,160,158,148,142,128,118,92,88,82,80,62,52,50,40,38,22,10,8 190 183,177,159,147,123,117,87,81,51,39,33,9 191 188,168,162,120,102,90,78,60,42,0 192 187,181,175,161,155,145,139,125,121,119,89,85,79,65,41,35,19,1 193 190,186,180,174,156,120,114,84,36,30,0 194 189,165,153,123,87,63,57,45,3 195 188,184,178,172,164,158,154,152,142,136,122,116,112,98,86,82,68,56,46, 44,38,32,28,16,4,2 196 193,183,177,153,135,117,45,33,15,3 197 192,186,156,150,114,96,84,66,60,30,0 198 191,185,181,175,169,161,155,151,139,119,115,109,95,71,59,41,35,31,25,1 199 180,168,138,132,72,42,0 200 197,189,183,159,153,147,117,111,93,69,63,51,33,27 201 196,188,182,178,172,158,148,130,112,92,70,62,50,38,22,10 202 199,195,171,165,135,129,105,75,39,21,9 203 198,186,180,156,150,144,114,90,66,54,36,30,24 204 197,193,185,175,163,145,143,133,107,103,77,73,67,65,53,47,37,25,23,7 205 192,174,168,162,144,132,126,108,102,66,24,6 206 203,195,183,177,153,147,105,75,57,33,27 207 202,194,190,176,166,160,146,140,124,110,106,104,100,70,56,50,44,34, 26,16 208 201,189,171,165,141,129,105,99,69,15 209 192,180,150,138,108,102,72,60,42,30,18 210 199,191,187,179,173,169,163,157,149,143,139,137,127,121,107,103,101,97, 83,71,61,59,53,47,31,29,19,17,13 211 208,198,168,138,102,60,30,18,12,0 212 209,207,189,171,141,105,99,81,45,39,21,15 213 208,206,196,184,176,170,166,160,154,146,140,134,124,104,100,64,56,50,20, 16,14 214 207,195,183,153,135,117,63,57,15 215 204,186,174,168,144,132,102,78,66,48,42,36,24,18 216 205,203,193,185,173,163,157,143,137,133,115,77,65,53,35,25,23,17 217 214,204,180,156,150,120,114,90,60,54,24,6 218 215,213,201,171,165,135,129,51,45,39,21

N N的素公差219 214,212,202,200,190,182,178,160,148,140,130,118,112,92,88,62,52,38,22, 20,8 220 213,201,189,177,159,153,147,117,111,93,57,21,9 221 218,210,198,180,168,162,138,132,90,72,48,30 222 217,211,209,199,179,175,161,151,125,115,109,95,91,85,71,59,55,49,29,11 223 220,216,210,186,156,150,144,126,114,84,0 224 219,207,195,177,165,135,123,93,87,57,45,33,27 225 218,214,208,206,196,194,184,172,164,158,154,142,128,124,122,112,88,86, 68,52,46,44,32,26,14,2 226 223,213,207,195,183,153,147,123,87,45,15,3 227 222,216,204,174,156,126,120,90,54,30,0 228 221,215,211,205,191,181,169,161,155,145,139,131,125,121,119,89,79,65,49, 29,5,1 229 210,192,168,150,120,102,78,48,0 230 227,219,213,201,189,171,159,129,123,117,81,63,51,39,3 231 226,218,212,208,202,200,190,188,178,170,158,152,148,142,128,118,100,82, 80,50,40,38,20,8,2 232 229,225,201,189,165,135,105,81,75,39,9 233 230,228,216,210,186,150,126,120,84,60,36,6,0 234 229,227,223,215,205,197,187,175,167,163,155,145,133,125,103,97,83,77,43, 37,35,23,7,5 235 232,228,222,204,198,174,162,138,132,96,78,72,42,36,6 236 231,225,213,207,195,183,165,153,147,123,57,45,3 237 230,226,224,220,206,196,194,184,164,136,130,110,100,80,74,70,56,44,40, 14,4 238 225,219,201,195,171,159,141,135,129,111,99,39 239 228,222,210,192,180,150,108,72,42,12,0 240 237,227,223,221,217,209,199,193,181,179,169,161,157,143,139,133,127,113, 109,91,77,67,41,29,17,11,1 241 238,236,222,198,180,168,138,132,90,30,0 242 237,235,225,219,189,159,141,111,105,75,69,51,15,9 243 236,224,220,214,206,196,190,176,154,146,140,136,130,116,106,104,94,64, 50,14 244 233,213,177,165,135,105,93,87,63,33 245 242,234,232,222,216,214,186,174,156,144,138,114,108,72,66,48,12,6 246 241,233,217,215,213,193,187,185,175,173,163,143,137,133,107,65,47,35, 23,5 247 244,240,230,216,210,186,174,150,120,90,84,24 248 243,231,229,219,211,141,135,111,99,69,21,9 249 242,238,230,218,212,208,190,182,170,160,152,148,140,118,110,100,98,82, 68,58,22,20,8 250 237,227,213,207,189,183,171,147,123,99,87,57,27,21 251 248,240,228,210,198,180,168,150,138,120,60,18,0 252 247,239,235,215,211,209,205,181,179,169,149,145,121,115,101,95,85,79,59, 55,41,29,25,19 253 250,246,234,224,210,186,180,156,144,126,114,96,54,24 254 249,237,225,223,213,207,195,165,147,105,63,57,27,15,3

N N的素公差255 248,244,236,232,224,212,208,202,194,184,176,166,154,146,142,128,124,118, 104,98,92,82,76,62,56,26,22,16,14 256 253,243,183,177,153,117,93,75,15 257 252,246,234,220,210,204,174,144,126,90,60,24,0 258 251,245,241,229,221,205,199,191,175,161,151,131,121,109,101,95,91,79,59, 35,19 259 240,228,218,198,180,162,150,120,108,78,48,18 260 249,243,231,219,217,207,201,189,171,159,129,123,93,87,33,21,9 261 248,242,238,230,218,202,200,188,178,172,160,158,149,122,112,98,88,70,50, 32,20,10 262 259,225,215,201,195,159,135,111,105,69 263 260,258,246,240,216,204,156,126,96,90,84,30,6 264 259,257,245,235,227,223,203,197,193,185,175,167,157,155,137,133,125,115, 85,83,73,67,53,17,7 265 258,234,222,212,198,192,168,156,114,108,102,84,72,66,42 266 255,243,237,225,213,195,183,165,153,135,117,93,87,27,15 267 256,254,236,220,200,196,194,166,164,154,130,116,100,86,70,44,40,26,10 268 255,231,209,195,189,171,165,141,129,111,105,69,39 269 252,240,210,208,198,180,162,132,120,90,78,42,12,0 270 267,253,251,239,233,229,217,209,197,191,187,169,163,161,139,131,119,113, 103,97,89,79,77,47,41,37 271 266,252,228,192,168,162,108,78,60,0 272 269,265,249,231,205,189,159,81,75,45,39,21 273 268,248,236,230,226,214,206,194,190,184,176,166,160,146,136,124,116,110, 106,100,94,80,76,74,44,34,4 274 267,263,213,203,165,147,153,123,93,75,63,33,3 275 272,262,246,234,228,204,202,192,168,174,144,126,108,84,78,42,36,18,6 276 271,265,247,245,233,223,203,187,173,163,145,125,113,103,97,83,77,35,5 277 270,264,260,246,210,180,120,96,54,0 278 259,231,225,199,189,171,141,111,105,81,39 279 268,262,242,220,208,200,182,178,170,152,142,140,130,122,100,88,80,68,52, 38,28,2 280 277,267,261,257,243,219,207,197,183,177,153,141,129,117,99,87,69,57,51 281 276,240,228,222,198,180,168,150,108,102,30,12,0 282 275,265,259,241,239,221,185,181,179,175,151,119,115,101,91,85,71,55,49, 25,11 283 280,264,254,240,216,204,194,180,174,156,126,90,84,54 284 279,273,253,237,225,195,183,177,147,135,117,105,33,27 285 278,272,262,256,238,224,218,214,202,182,178,176,172,154,148,146,136,134, 112,104,94,88,74,62,52,46,8 286 283,255,213,177,147,153,123,93,87,63,45 287 284,282,276,270,234,216,190,180,174,114,96,60,30,6 288 283,281,275,269,259,235,221,215,199,191,179,175,161,151,131,121,109,95, 91,65,61,59,49,19 289 282,258,252,248,210,188,150,132,108,90,78,60,48,18 290 287,279,273,267,247,231,219,189,187,177,159,141,111,99,93,63,57,21 291 286,280,278,272,250,232,230,218,212,208,188,142,140,128,118,110,98,92, 68,62,58,40,20

N N的素公差292 285,279,255,249,245,231,195,185,165,141,129,81,15 293 276,270,264,210,186,184,156,126,96,66,60,54,24,0 294 283,277,275,263,253,247,227,215,205,193,185,167,163,155,145,137,127,115, 103,95,65,55,53,43,37,23,17,13 295 292,282,276,252,242,228,192,182,168,144,138,114,102,84,72,54 296 291,273,267,225,213,207,183,165,123,105,63,57,15 297 290,280,274,266,260,250,244,226,224,190,170,166,134,124,104,100,86,70, 56,40,20 298 295,279,239,225,201,189,159,141,135,99,75,69 299 294,288,270,258,238,210,168,162,150,132,120,102,60,48,18 300 293,287,277,271,269,263,257,247,241,221,203,199,187,163,161,149,133,121, 119,109,101,89,73,67,59,49,31,7 301 298,270,240,222,198,162,138,120,108,78,72,30 302 299,297,291,285,261,255,235,219,201,175,165,129,51,45,9 303 298,296,290,284,274,266,260,244,220,206,200,196,176,164,154,146,136,130, 106,80,76,70,64,46,34 304 297,273,267,243,237,233,195,173,153,105,93,75,63,33,27 305 302,294,288,282,264,258,252,232,216,204,198,174,156,126,114,78,54,48,12 306 301,295,293,287,265,263,235,217,203,197,193,157,155,143,133,127,125,115, 113,95,83,77,73,67,25 307 300,294,270,264,240,234,180,170,156,150,126,114,66,30,0 308 291,285,279,261,255,249,229,201,195,171,169,159,141,111,81,75,51,39 309 298,292,290,278,268,262,248,238,212,200,178,170,158,152,130,112,110,80, 70,58,40,38,28,2 310 307,297,291,267,237,231,227,213,153,147,129,111,99,87,69,39 311 308,306,288,282,258,252,210,198,138,120,78,72,42,0 312 307,305,295,289,281,275,265,259,251,245,229,211,209,175,155,149,145,121, 119,89,85,71,61,55,41,35,19,5 313 306,294,234,224,210,186,174,164,150,120,84,36 314 303,285,273,255,243,207,165,163,147,135,117,87,75,45,33,3 315 304,302,292,286,284,278,272,262,256,254,248,242,232,226,208,206,188,184, 164,152,148,142,134,124,118,116,104,86,82,74,64,58,44,38,34,22 316 303,285,255,207,123,117,105,93 317 314,300,276,270,246,220,204,186,160,150,144,84,66,36,0 318 313,301,299,289,281,275,259,251,245,239,229,205,191,181,169,161,145,139, 121,91,79,61,49,41 319 312,300,288,282,258,252,222,218,180,168,138,120,90,78,48,12 320 297,279,273,267,249,237,217,189,183,157,147,141,129,81,69,63,39,27 321 310,298,280,278,250,248,242,220,182,158,142,140,128,110,98,80,52,28,10 322 319,309,285,279,255,249,225,219,215,165,155,141,111,99,51,45,15 323 320,318,294,276,270,264,240,224,214,186,156,144,126,96,66,30 324 319,317,307,295,293,283,277,263,253,245,223,217,197,185,175,143,133,125, 97,85,73,55,43,13,7

N N的素公差325 322,318,306,294,282,252,246,222,198,174,162,152,132,114,96,84,54,48 326 321,315,273,267,243,237,195,177,153,135,93,75,57,33 327 320,316,314,304,290,280,274,266,260,244,230,220,214,196,176,160,136,134, 130,104,94,70,56,46,20,10 328 325,315,291,249,219,171,149,135,129,105,51,21 329 324,318,312,288,270,258,240,228,192,180,150,148,132,102,90,72,60,18 330 323,317,313,311,301,289,287,277,271,269,263,257,247,241,233,227,217,193, 191,179,173,157,149,137,133,131,119,103,101,91,89,79,59,53,49,37,23,19 331 328,312,300,288,270,192,168,132,108,102,90,0 332 329,327,321,315,309,285,261,231,225,205,135,99,51,21,15 333 328,326,320,314,310,286,274,266,260,254,244,236,230,224,176,170,166,154, 134,106,100,76,56,40,26,16 334 327,297,273,267,237,207,203,153,143,123,105,63,3 335 324,318,312,306,282,264,252,234,228,222,186,168,144,142,96,84,66,54, 24,18 336 325,323,317,307,305,295,283,265,263,257,235,233,227,205,187,185,173,163, 143,125,113,103,97,95,85,65,43 337 324,306,294,270,264,240,234,210,200,186,140,126,96,60,30,0 338 335,321,315,309,279,255,249,231,225,199,171,165,141,139,111,81,45,21 339 334,332,320,308,302,292,280,278,268,260,238,232,230,208,202,182,160,148, 140,128,110,100,82,70,62,58,28,8 340 337,333,321,303,279,267,261,237,231,201,183,159,147,117,99,69,9 341 336,318,312,300,258,252,228,168,162,108,90,60,48 342 335,331,319,311,305,301,299,289,275,259,245,235,229,215,179,161,145,119, 115,91,25,11,5 343 340,330,300,276,264,234,204,194,180,144,120,114,66,36,6 344 339,333,315,303,297,273,255,243,213,193,177,165,133,117,105,87,75 345 338,332,328,326,314,308,302,298,286,274,272,262,256,248,242,232,218,196, 178,164,154,134,122,118,116,112,104,94,88,76,74,64,52,38,34,14,8 346 327,315,285,273,195,153,117,75 347 344,336,330,306,300,294,246,240,216,210,190,174,156,120,114,54,36,0 348 343,335,329,325,311,305,295,269,259,251,245,239,221,209,199,175,155,151, 119,115,109,91,71,41,31,11,1 349 342,312,282,270,252,222,198,192,150,138,108,72,18,0 350 333,327,309,303,297,291,267,249,243,237,219,213,187,171,159,153,127,117, 111,99,81,69,39,33,3 351 340,332,322,310,308,292,290,280,268,250,248,242,220,212,172,170,158,152, 128,112,110,82,80,70,58,2 352 349,339,321,309,291,279,255,249,225,195,189,185,171,125,111,81,45,15 353 348,330,324,306,300,294,264,246,240,216,204,156,126,124,114,96,36,6,0 354 347,337,323,307,293,287,265,253,247,245,223,217,215,203,187,155,125,113, 103,85,77,47,43,5 355 336,318,288,276,252,246,216,192,182,144,132,122,84,78,24

N N的素公差356 353,345,327,303,297,285,243,207,165,123,105,75,63,45,3 357 352,344,334,326,320,316,304,296,290,286,284,274,260,250,244,230,220,206, 200,190,184,166,164,146,130,106,100,76,64,40,26,10 358 351,315,285,261,249,219,179,165,129,119,81,51,21,9 359 342,318,300,288,258,228,210,178,162,120,118,108,102,90,42,0 360 349,341,331,323,317,313,301,299,293,287,281,271,259,257,247,233,211,209, 203,197,187,181,163,161,149,127,119,103,89,79,49,29,23,7 361 358,348,330,300,282,258,210,180,162,138 362 357,339,321,315,291,279,255,231,225,195,105,69 363 356,346,320,310,296,290,284,280,256,254,236,224,214,206,200,184,140,136, 124,94,86,70,56,46,26,16,4 364 345,327,297,267,255,237,213,207,183,173,135,123,113,93,57,33,15 365 362,354,336,318,312,294,282,276,252,234,228,198,192,172,138,114,96,84, 54,18 366 361,353,343,335,325,307,295,293,287,277,265,253,235,227,203,175,155,143, 137,133,97,95,73,55,35,17,13 367 360,324,306,294,264,240,210,204,174,170,156,110,96,90,30,0 368 365,351,315,309,285,279,231,219,201,195,189,169,141,135,111,99,51,21,15 369 364,358,350,340,332,322,308,290,272,262,238,232,230,218,202,188,178,172, 140,130,118,98,92,88,62,52,32,20,10 370 363,357,339,303,291,273,261,231,207,177,171,129,93,63,39 371 368,348,330,312,288,282,270,222,198,192,138,132,90,78,60,18,12 372 367,361,355,329,319,311,305,301,289,275,271,269,259,245,235,221,215,205, 199,191,175,149,131,115,95,91,61,25 373 370,366,360,354,336,300,270,246,234,174,150,104,66,36,24,6 374 369,345,327,303,285,273,267,243,225,195,183,163,147,135,105,103,93,57, 27,15 375 368,364,358,352,344,334,316,308,302,286,278,272,268,266,244,226,224,218, 212,202,196,194,182,148,146,134,124,104,82,64,58,44,26,22,8 376 363,357,333,315,297,267,225,195,165,147,45 377 374,366,324,306,276,270,264,240,210,186,180,144,126,100,84,24 378 373,365,361,355,349,341,331,305,299,295,281,275,269,265,241,239,229,221, 215,199,185,179,145,121,109,101,85,71,61,41,31,19,11 379 372,360,348,312,282,252,240,228,222,198,168,108,98,42,30,0 380 377,363,339,321,297,279,273,267,213,207,189,183,157,141,129,123,99,87, 69,21 381 376,370,362,358,352,338,328,320,310,302,292,280,278,272,250,218,190,188, 182,142,140,50,28,2 382 379,375,369,351,345,309,279,225,219,189,159,155,141,105,75,51,15 383 378,360,336,300,294,276,270,234,216,210,204,186,154,126,66,36,0 384 377,373,367,355,343,325,317,277,275,257,247,235,233,217,203,193,187,185, 173,157,115,103,77,73,47,37,35,25,17,5

N N的素公差385 372,366,354,348,342,324,306,288,276,258,246,234,222,192,186,162,156,152, 114,92,78,54,48,36 386 383,375,357,333,315,297,273,255,213,207,135,117,93,75,33,3 387 382,374,370,364,356,346,340,314,304,290,286,274,260,256,230,220,214,206, 190,176,160,154,136,116,80,76,70,34 388 385,381,369,351,345,321,285,231,189,159,149,111,51,21,9 389 384,372,330,288,258,252,210,198,148,132,120,78,72,42,30,0 390 383,379,371,367,361,353,349,343,337,329,319,311,301,293,287,283,263,253, 251,241,227,217,211,209,197,179,167,157,151,133,119,113,109,97,73,59,43, 41,31,11,7 391 378,360,348,318,282,252,240,228,210,180,150,42 392 381,369,351,309,291,285,261,255,225,201,195,165,111,85,75,39,9 393 380,376,364,350,346,340,334,326,290,284,280,266,254,226,214,200,194,170, 164,154,116,86,56,46,40,26,4 394 375,363,357,333,315,297,267,237,213,183,153,83,63,45,27,15 395 392,378,366,348,324,306,288,282,264,258,246,222,204,198,168,162,126,114, 84,36,6 396 391,377,373,365,355,343,337,323,313,295,287,265,257,255,247,233,205,203, 197,173,167,145,127,125,103,65,43,37,13 397 390,360,354,336,330,294,246,234,204,174,140,126,90,80,66,60,0 398 395,375,345,285,261,249,219,201,171,165,159,105,81,51 399 394,388,370,362,358,352,340,328,320,310,302,292,262,260,248,242,232,220, 218,208,202,200,188,172,170,158,148,142,122,88,68,62,50,40,32,20,10,2 400 397,393,387,369,357,339,333,327,291,273,261,243,219,207,201,177,171,123, 63,33,21 401 396,372,360,342,318,300,252,168,162,120,108,48,18,0 402 395,391,385,371,359,355,349,341,331,299,289,275,271,251,245,239,229,205, 191,175,169,161,145,121,85,65 403 390,384,366,336,330,324,306,240,204,174,134,96,54,36,6 404 393,357,315,297,273,255,237,213,165,153,133,73,57,45,15 405 392,388,382,368,364,352,346,338,334,322,304,296,278,268,256,254,248,242, 238,226,214,212,194,182,172,166,164,136,98,94,74,58,56,52,26,16 406 403,378,363,345,333,327,303,267,255,225,213,195,165,135,57,27 407 404,402,390,366,354,336,294,276,270,240,234,210,180,156,150,130,114,96, 70,60,54,24 408 403,401,389,385,379,365,361,349,335,325,319,311,301,269,251,245,235,211, 209,185,179,169,139,115,101,91,71,59,55,49,41,25,11 409 402,378,348,342,330,300,282,252,210,198,168,138,132,128,78,30,12,0 410 399,387,363,351,309,273,243,237,231,183,177,159,153,99,93,57,51,21 411 400,398,382,358,350,340,332,328,322,308,298,280,272,262,248,232,230,220, 188,182,160,130,52,28,22,10 412 409,399,381,375,345,339,315,261,249,231,219,189,135,75,65,45

N N的素公差413 410,408,396,384,360,330,306,264,246,240,234,186,180,174,156,144,96,66, 64,54 414 409,407,397,395,383,373,355,347,343,325,312,305,287,277,263,247,233,217, 187,185,173,163,157,143,133,107,65,47,35,25,17,5 415 412,408,396,378,372,354,342,336,318,312,276,258,216,204,192,186,122,108, 84,62,48,6 416 413,411,405,393,357,345,327,303,285,267,243,237,225,183,177,147,105,63, 33,15 417 412,410,406,404,394,380,376,370,356,344,334,316,310,266,244,236,226,224, 190,184,176,160,146,140,124,106,86,70,50 418 411,405,375,351,339,321,315,309,291,255,245,189,81,69,59,39,21 419 408,402,390,378,282,240,228,222,180,168,150,138,102,60,30,0 420 407,403,401,391,389,377,373,367,353,349,341,337,331,319,313,307,289,281, 271,263,257,253,241,239,227,223,221,197,187,181,179,149,143,127,103,83, 67,41,37,19,11,1 421 418,402,390,348,318,312,270,240,222,210,198,180,42,10,0

[1]华罗庚 .数论导引 [M].北京:科学出版社,1957.

[2]陈景润 .初等数论 [M].北京:科学出版社,1988.

(责任编辑 刘洪基)

The Proof of Goldbach Problem

NI Jia-ta i
(The Fifteenth M iddle School of Panzhihua City,Panzhihua617063,China)

The Goldbach conjecture is the world famous problem,more than 100 years,it had not been proved.Now still arguing,can we prove thisproblem using elementary number theory?The author proves the famous problems successfully.We can see that the scientific debate about Goldbach conjecture had been subsided.In this paper,based on the natural numberN >2 as the arithmetic mean,the author seeks the prime number common differenced.For the arithmetic mean 3≤N≤421,the prime number common difference had been enumerated one by one,and if the arithmetic mean N >421,the author then proved the existence of its prime number common difference. The problem of prime twins if it belongs to Goldbach problem,the author also proved it.

Goldbach problem;prime number common difference;congruence;Chinese Remainder Theorem;twin primes

O156

A

1671-7406(2010)03-0001-17

2010-01-04

倪家泰 (1941—),男,云南楚雄,中学高级教师 (已退休),主要研究方向:数论。