倪家泰

(攀枝花市第十五中學(xué),四川 攀枝花 617063)

關(guān)于 Coldbach問(wèn)題的證明

倪家泰

(攀枝花市第十五中學(xué),四川 攀枝花 617063)

Coldbach問(wèn)題是著名的世界數(shù)學(xué)難題,一百多年以來(lái)未曾有人能證明它。用初等數(shù)論能不能證明這一問(wèn)題,現(xiàn)在也爭(zhēng)論不休。我的論文成功的證明了這一問(wèn)題,并且解決了這個(gè)爭(zhēng)論。

本文以自然數(shù) N>2為等差中項(xiàng),尋求素公差 d。對(duì)于 3≤N≤421的等差中項(xiàng),一一列舉其素公差,對(duì)于N>421的等差中項(xiàng),證明其素公差存在。孿生素?cái)?shù)對(duì)問(wèn)題是哥氏問(wèn)題的特殊情況,同時(shí)進(jìn)行證明。

Coldbach問(wèn)題;素公差;同余式;孫子定理;孿生素?cái)?shù)

定義 1 自然數(shù)N>2,若存在非負(fù)數(shù) d為公差,使N-d和N+d為二素?cái)?shù),則稱 d為N之素公差。

定義2 稱同余式

n≡r(r與 N和 -N不同余)modq(0≤r

N關(guān)于 2的補(bǔ)同余式有且僅有一個(gè)。若 q|N,N關(guān)于 q的補(bǔ)同余式有 q-1個(gè):若 q不能整除N,N關(guān)于 q的補(bǔ)同余式有 q-2個(gè)。

當(dāng) d〈N-qk時(shí),d和 ±N對(duì)所有小于全體素?cái)?shù)都不同余,故得到下列同余式方程組:

稱此方程組為N的補(bǔ)同余式組。

由孫子定理,N的每一補(bǔ)同余式組在 0到 q1q2……qk間有且僅有一解 dN。

dN為 N的補(bǔ)同余式組 (1)之解,dN就滿足方程中的每一同余式,dN滿足第 i(i=1, 2……k)個(gè)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的因數(shù),也不是N+dN的因數(shù),換言之素?cái)?shù)列 (Ⅰ)中的任何一個(gè)素?cái)?shù)既不能整除 N-dN,也不能整除N+dN,因而N-dN和N+dN都是素?cái)?shù)。

是N的補(bǔ)同余式組 (1)之從小到大排列的解。

定義 3 令 b1=c1,bk-bk-1=ck稱 ck為 N的補(bǔ)同余式組 (1)之解差。解差中的最大者稱N補(bǔ)同余式組之最大解差,簡(jiǎn)稱N的最大解差。

例 1 求N=62時(shí)同余式組 (1)之全體解及其最大解差和平均解差?

∴素?cái)?shù)列 (Ⅰ) =2,3,5,7,11

62關(guān)于 2,3,5,7,11的余數(shù)分別 0,2,2,6,7。62的補(bǔ)同余式組 (1)為

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

其解從小到大排列如下:

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。

其最大解差為 48。

引理 2 當(dāng)N>420時(shí),數(shù)N的補(bǔ)同余式組 (1)之最大解差小

證明:數(shù)N的補(bǔ)同余式組之解必須滿足前二個(gè)同余式

n≡r1(r1與N不同余) mod q1

n≡r2(r2與N和 -N不同余) mod q2

若這兩個(gè)同余式構(gòu)成的同余式組之解為 a(當(dāng) 3不能整除N時(shí),r2=0,a的值有且僅有一個(gè);當(dāng) 3|N,r2可為 1或 2,a的值有二個(gè)),則 N的補(bǔ)同余式組之解具有 a+6m的形式。

對(duì)于 a的一個(gè)固定值,N的補(bǔ)同余式組解中,具有 a+6m形式之解的最大解差中不是解的數(shù)之個(gè)數(shù),是m從某自然數(shù)起連續(xù)幾個(gè)自然數(shù)的個(gè)數(shù)。這些數(shù)必然是滿足下同余式中至少一個(gè)同余式。

由于 (2)中同余式個(gè)數(shù)最多有 2(k-2)個(gè),不重復(fù)滿足 (2)中同余式之一的情況下,這些第一次滿足 (2)中同余式之?dāng)?shù)在從起不是解的數(shù)中,不妨記這些數(shù)從小到大排列為

為保證 (2)中同余式的第一次滿足最多,我們約定 (3)中若(1≤i≤l)同時(shí)為qt和 qr(3≤t

在此基礎(chǔ)上研究重復(fù)滿足 (2)中同余式之一的情況個(gè)數(shù)。重復(fù)滿足 (2)中同余式之一的數(shù)只需插入其中間,插入情況及個(gè)數(shù)為:若(1≤j≤l)滿足 (2)中同余式 n≡rimod qi,當(dāng) qi|N時(shí),第一個(gè)重復(fù)滿足數(shù)前有 qi個(gè) (2)中同余式滿足數(shù)。其余每隔 qi-1個(gè) (2)中同余式滿足數(shù)有 1個(gè)重復(fù)滿足數(shù):當(dāng) qi不能整除N時(shí),第一個(gè)重復(fù)滿足數(shù)前有 qi個(gè) (2)中同余式滿足數(shù),其余每隔 qi-2個(gè) (2)中同余式滿足數(shù)有 2個(gè)重復(fù)滿足數(shù)。顯然 qi|N時(shí)的重復(fù)滿足數(shù)個(gè)數(shù)沒(méi)有 qi不能整除N時(shí)多,所以計(jì)算最大解差中重復(fù)滿足數(shù)個(gè)數(shù)時(shí)均以 qi不能整除N的情況來(lái)估計(jì) (2)中所有同余式的重復(fù)滿足個(gè)數(shù)。

為了使最大解差估計(jì)中重復(fù)滿足數(shù)最多,我們約定,每次計(jì)算時(shí)若末尾的個(gè)數(shù)在 1到qi-3內(nèi),不論多少均估計(jì)有一個(gè)重復(fù)滿足數(shù)。且依下方法步驟計(jì)算:假設(shè)滿足 (2)中第一個(gè)同余式,重復(fù)滿足數(shù)個(gè)數(shù)計(jì)算得 l1,插入這 l1個(gè)數(shù)后排列 (3)有 l+l1個(gè)新排列數(shù),再設(shè)滿足 (2)中第二個(gè)同余式,因和第二次的重復(fù)滿足數(shù)之計(jì)算無(wú)關(guān),故以1個(gè)排列數(shù)來(lái)計(jì)算其重復(fù)滿足數(shù)個(gè)數(shù),……像這樣依次計(jì)算 (2)中各同余式之重復(fù)滿足數(shù)個(gè)數(shù),并在此基礎(chǔ)上適當(dāng)調(diào)整順序以使重復(fù)滿足數(shù)個(gè)數(shù)達(dá)到最多。

例 2 求 k=30時(shí),(2)中各同余式重復(fù)滿足數(shù)個(gè)數(shù)和相應(yīng)的最大解差估計(jì)值?

解 k=30,qk=113,2(k-2) =56, (2)中各同余式重復(fù)滿足數(shù)計(jì)算情況及個(gè)數(shù)如下:

計(jì)算式 個(gè)數(shù) 計(jì)算式 個(gè)數(shù)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)中各同余式滿足數(shù)為 431+8+27=466

記N補(bǔ)同余式之解的最大解差中滿足 (2)中同余式的數(shù)之個(gè)數(shù)為 f(k),則最大解差為 6[f(k) +1]。按上述方法計(jì)算不難得到下表:

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

上表中說(shuō)明引理對(duì) 10≤k≤34時(shí)相應(yīng)的N成立的,并且每當(dāng) k增加 1時(shí),最大解差中滿足 (2)中同余式之?dāng)?shù)的個(gè)數(shù)增加數(shù)不大于 k+12。如 k從 32增加到 33時(shí)最大解差中滿足 (2)中同余式之?dāng)?shù)的個(gè)數(shù)增加了

事實(shí)上,當(dāng) k每增加 1,排列 (3)中的初次滿足數(shù)最多增加 2個(gè),這 2個(gè)數(shù)最多使(2)中關(guān)于 q3到 q9的相應(yīng)各同余式平均增加 2個(gè)重復(fù)滿足數(shù),合起來(lái)共同增加 16個(gè)滿足(2)中的數(shù)。這 16個(gè)數(shù)又最多使 (2)中關(guān)于 q10到 q19的相應(yīng)各同余式平均增加一個(gè)重復(fù)滿足數(shù),這時(shí)最大解差中滿足 (2)中的數(shù)一共最多增加 26個(gè)數(shù),故 (2)中不增加重復(fù)滿足數(shù)的同余式越到后面越多,其增加的重復(fù)滿足數(shù)個(gè)數(shù)平均不到 1個(gè)。所以 (2)中前 k-2個(gè)同余式在最大解差中最多增加 k+6個(gè)滿足 (2)中同余式之?dāng)?shù)。

k增加 1時(shí),(2)中同余式個(gè)數(shù)也增加一個(gè)20),所以 (2)中后面 qi(i≥20)的相應(yīng)同余式將有不增加重復(fù)滿足數(shù)情況出現(xiàn),由于即 k+1從 32起至少連續(xù)增加 3個(gè)值,上同余式的重復(fù)滿足數(shù)才可能增加2個(gè)數(shù),但這時(shí) (2)中不增加重復(fù)滿足數(shù)的同余式也至少增加 2個(gè)。當(dāng) k+1越大, (2)中不增加重復(fù)滿足數(shù)的同余式就越多。

因此,當(dāng) k增加 1時(shí),最大解差中最多增加 (k+6) +6=k+12個(gè)滿足 (2)中同余式之?dāng)?shù)。即 f(k+1) -f(k)≤k+12當(dāng) k增加 1時(shí),素公差的取值范圍也從增大到其范圍擴(kuò)大了

這個(gè)同余式的重復(fù)滿足數(shù)個(gè)數(shù)從 k+1為 21起為 6。

由于 (2)中前 k-2個(gè)同余式在最大解差中增加最多 k+6個(gè)滿足 (2)中同余式之?dāng)?shù), k+6和 qk+1-2比較,當(dāng) k越大,其比值越小。如 k=31,則 k+6=37,而 qk+1-2=131-

要證 2qk+2≥6(k+12),只需證 qk≥3k+35,不難驗(yàn)證:上式從 k=32,qk=131起成立。所以引理對(duì)N>420的全體自然數(shù)成立。

若N的補(bǔ)同余式組中的解之 a取二個(gè)值,則其解為兩組形式如 a+bm的數(shù),其一組解之最大解差小另一組解的數(shù),無(wú)論在或不在這個(gè)最大解差之中,都不擴(kuò)大最大解差,故引理成立。

定理 1 大于等于 6的偶數(shù)可表示為兩奇素?cái)?shù)之和。

證明:只需證大于等于 3的自然數(shù)都有素公差。

后面附表已列出 2

當(dāng)N>420時(shí),由引理 2,N的同余式組之解的最大解差小于設(shè) N的補(bǔ)同余式組之解首先出現(xiàn)的最大解差由解 d和 d′構(gòu)成,d

若 d

若 d

令Ci表示 (2)中取值個(gè)數(shù)。由于 N的補(bǔ)同余式組之解不大于 q1q1q3……qk,所以全體解差和小于q1q2q2……qk,其平均解差小于N的素公差個(gè)數(shù)大約有

定理2 孿生素?cái)?shù)對(duì)有無(wú)窮對(duì)。

證明:自然數(shù)N≥6時(shí),若N有素公差 1,顯然N-1和N+1是一對(duì)孿生素?cái)?shù),這時(shí)N是下列同余式組

由于 k是大于等于 2的任意自然數(shù),故孿生素?cái)?shù)對(duì)有無(wú)窮對(duì)。

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]華羅庚 .數(shù)論導(dǎo)引 [M].北京:科學(xué)出版社,1957.

[2]陳景潤(rùn) .初等數(shù)論 [M].北京:科學(xué)出版社,1988.

(責(zé)任編輯 劉洪基)

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,中學(xué)高級(jí)教師 (已退休),主要研究方向:數(shù)論。