双生素数无上界
孪生素数猜想证明简述
孪生素数猜想证明简述一：逻辑证明（最简单，但逻辑思维要求高）根据素数新定义：从祖素数2开始，素数倍数后不连续的数即为素数。
易知素数除了2以外全是奇数，所以在奇数数轴上研究素数会有奇效。
奇数数轴：3,5,7,9,11,13,15,17,19,21,23,25,27,29,31......，无数对相差为2（相连）的数；假设只有3为素数，去掉其倍数后数轴变为：3,5,7,11,13,17,19,23,25,29,31......，只少了一点，但依旧有无穷对素数相差2；添加5为素数，去掉其倍数后数轴变为3,5,7,11,13,17,19,23,29,31......，少的更少，剩下相差为2的素数对肯定是无穷多；等等；如此可以无穷下去，但少的越来越少，而且剩余差值为2的素数对肯定是无穷多。
所以孪生素数肯定是无穷多的。
一目了然！！！当然也很容易看出，P和P+2k的素数对也是无穷多的（波利尼亚克猜想成立）。
（参考文献：奇数轴中素数量与合数宽度的研究）二：公式证明（难度极大）在上述的逻辑证明中，我们若将奇数数轴设为单位1；则3的倍数占比为：1/35的倍数占比为：1/5-1/157的倍数占比为：1/7-1/21-1/35+1/105等等，最后可得到孪生素数在奇数中的占比(LiKe级数公式)约为：1-1/3-(1/5-1/15)-(1/7-1/21-1/35+1/105)-(1/11-1/3*11-1/5*11-...+...)-...=1-1/3-1/5-1/7-......-1/p+1/15+1/21+......+1/pq-1/105-1/165-......-1/pqr+...-...=1-∑1/P+∑1/pq-∑1/pqr+…±∑1/∏P （1）(式中所有素数为奇素数，分母为偶数个素数积时取和，为奇数时取差)关于该新颖级数的求和不在此演示。
不过它是发散的(其值应该不为0)，该级数本身足以说明了孪生素数的无穷多。
中国剩余定理·双生素数猜想·哈代-李特伍德猜想(B)
中国剩余定理·双生素数猜想·哈代－李特伍德猜想(B)Chinese remainder theorem,twin primes conjecture andHardy-Littlewood conjecture(B)童 信 平**********************摘要 双生素数(p,p+k)中，①若p 、k 对模p 的余数相加等于p i i ，则p 不是双生素数(p,p+k)的答案；若i=1～r 时，p 、k 对模p 的余数相加皆不等于p i i ，则p 是双生素数(p,p+k)的答案。
②对于模p ～p 1r ，根据中国剩余定理，可以通过建立不同的同余式组，计算出双生素数(p,p+k)中的p 。
③(p+k)中必有素数，双生素数猜想成立。
同时指出，双生素数(p,p+k)有无限多对。
④讨论了哈代－李特伍德猜想(B )中的细节(未揭示的参变量)问题。
关键词 同余 中国剩余定理 双生素数猜想 哈代－李特伍德猜想(B)存在无限多对双生素数。
――这就是常说的双生素数猜想。
哈代－李特伍德猜想(B)则是关于双生素数(p,p+k)数量的猜想。
对于偶数k=2n 、不大于偶数(N -k)的素数p 和p ，我们有(p ,p +k)和(p,p+k)组成的正整数对， i i i ，(p (p )+k(p ①若p 、k 对模p 的余数相加等于p )=p i i i i i 。
)此时，p 不是双生素数(p,p+k)的答案；若i=1～r 时，p 、k 对模p ，(p (p )+k(p 的余数相加皆不等于p )≠p i i i i i 。
)此时，p 是双生素数(p,p+k)的答案。
②对于模p ～p 1r ，根据中国剩余定理，可以通过建立不同的同余式组，计算出双生素数(p,p+k)中的p 。
③(p+k)中必有素数，双生素数猜想成立。
同时指出，双生素数(p,p+k)有无限多对。
④讨论了哈代－李特伍德猜想(B )中的细节(未揭示的参变量)问题。
孪生素数
孪生素数要介绍孪生素数，首先当然要说一说素数这个概念。
素数是除了1 和它本身之外没有其它因子的自然数。
素数是数论中最纯粹、最令人着迷的概念。
除了 2 之外，所有素数都是奇数(因为否则的话除了 1 和它本身之外还有一个因子2，从而不满足素数的定义)，因此很明显大于2 的两个相邻素数之间的最小可能间隔是2。
所谓孪生素数指的就是这种间隔为2 的相邻素数，它们之间的距离已经近得不能再近了，就象孪生兄弟一样。
最小的孪生素数是(3, 5)，在100 以内的孪生素数还有(5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) 和(71,73)，总计有8组。
但是随着数字的增大，孪生素数的分布变得越来越稀疏，寻找孪生素数也变得越来越困难。
那么会不会在超过某个界限之后就再也不存在孪生素数了呢？我们知道，素数本身的分布也是随着数字的增大而越来越稀疏，不过幸运的是早在古希腊时代，Euclid 就证明了素数有无穷多个(否则今天许多数论学家就得另谋生路)。
长期以来人们猜测孪生素数也有无穷多组，这就是与Goldbach猜想齐名、集令人惊异的简单表述和令人惊异的复杂证明于一身的著名猜想- 孪生素数猜想：孪生素数猜想：存在无穷多个素数p, 使得p+2 也是素数。
究竟谁最早明确提出这一猜想我没有考证过，但是一八四九年法国数学Alphonse de Polignac 提出猜想：对于任何偶数2k,存在无穷多组以2k 为间隔的素数。
对于k=1，这就是孪生素数猜想，因此人们有时把Alphonse de Polignac作为孪生素数猜想的提出者。
不同的k 对应的素数对的命名也很有趣，k=1 我们已经知道叫做孪生素数，k=2 (即间隔为4) 的素数对被称为cousin prime (比twin 远一点)，而k=3 (即间隔为6) 的素数对竟然被称为sexy prime (这回该相信“书中自有颜如玉”了)！不过别想歪了，之所以称为sexy prime 其实是因为sex 正好是拉丁文中的6。
孪生素数无限!(格点数论版)
当2≥n 时, 在开区间))1(2,(211-='+-n n np p A 内至少有一对孪生素数. ------孪生素数无限(格点数论版).张 忠（言）江苏省南通市崇川区 邮编：226002摘要: 本文依据同余理论, 通过格点二次筛法对联立一元二次不同余方程组的解集:22{nn n n b -=∆δδ),(mod 0i p ,2,1=i ….n .}Z b ∈ 的分析与验证, 发现整数的一个重要规律:在前闭后开区间))1(2,0[21-=-n n p A 内至少有一个n n ∆∈δ. 依据该规律, 本文证明了“孪生素数”无限.关键词: 素数, 整数的多维式, 模, 不同余, 格点筛法, 集合的势.0. 引言.大于4的偶数是否都可表为二个奇素数之和? 孪生素数对是否无限? 这些都是一直困惑着人们的古老数论问题, 甚至许多大数学家都认为: 人们至今也未能找到真正能解决这些问题的方法和途径. 而本文谨用同余理论和筛法, 来揭示至少可解决“孪生素数对无限”的一个重要规律.1. 基本慨念, 名词, 定义及代(符)号的意义.1.1. 若无特别声明, 本文中小写字母表整数, 大写字母表整数集合. 例:n π=21p p …n p , )(n πφ表n π的欧拉数, }1),({==Ξn n n n πξξ表模n π的简化剩余集,P 表素数集合, 且P{b b p b P 2)()(=＜)(b p ＜b 2且}Pb p b ∈-)(2)(b P '.1.2. A 表集合A 的势, 即集合A 内元素的个数. 1.3. ≡ 为同余符号, 为不同余符号.1.4. 整数b 的多维式. 若⎪⎩⎪⎨⎧≡)(mod )(mod )(mod 2211n n p r p r p r b , 则可将其记作: ,,{21r r b ≡….}n r ()n πmod ,并称其为b 的多维式,在不至引起误解时,可省略式中()n πmod . 而由孙子定理与欧拉定理知:()i p inni i pr b φπ⎪⎪⎭⎫⎝⎛≡∑=1()n πmod1.5. 定义一: 定义一元一次不同余方程)(k k rx k r ()k p mod ,k r ≤0(＜)1,≥k p k , 为(素数)模k p 之k r 的(一次)筛, 简记为)(k k r s , 例: )2(33s 为: 而该不同余方程的解称)(k k r x 称为)(k k r s 的缩剩余, )(k k r X 为)(k k r s 的缩剩余集. 作为特例, 当)(k k rx 0=k r ()k p mod 时, )0(k x 称为模k p 的简化剩余,)0(k X 为模k p 的简化剩余集.1.6. 定义二. 若: n n b π≤≤1, ,,{21r r b n ≡….}n r , 则定义联立(一次)不同余方程组:xi n r b ≡()i p mod ,i r ≤0(＜)i p ,,2,1=i ….n 为(合数)模n π之n b 筛, 并简记为: )(n n b S 或,,({21r r S n ….})n r . 该联立方程的解x 称为(合数)模n π之n b 筛的(一次)缩剩余. 作为特例: 当0=n b 时, 该联立方程的解x 即模n π的简化剩余}1),({==Ξ∈n n n n n πξξξ. 图一: )0(.})0,0,0({)0(132133s S S ≡≡)0(2s )0(3s 的格点筛紧 接 下图紧 接 上 图由图一: )0(3S 可得模305323=⨯⨯=π的最小正简化剩余系: .}29,23,19,17,13,11,7,1{min 3=Ξ⊕. 1.7. 定义三: 若k r b ≡()k p mod ,)(201k k p r φ-≤≤,1＜Nk ∈,则定义不同余方程:kx k r b ±≡±()k p mod ,为(素)模k p 之)(b ±(或k r ±)的二次筛, 并简记为)(k k r s ±; k x 为)(k k r s ±的二次缩剩余, 为区别k x 与模k p 之其它二次筛的缩剩余, 模k p 之)(k r ±筛的缩剩余记为)(b x k ±或)(k k r x ±.因当2≥k 且kr b ≡0()k p mod 时: b 与b -分别为模k p 的两个不同剩余类, 但模k p 之b的二次筛与模k p 之b -的二次筛相同, 故模k p 之b 的二次筛与模k p 之b -的二次筛为模k p 之异名同类筛, 故知模k p 之二次异名同类筛的二次缩剩余也相同. 模k p 之)(k r ±筛系内有且仅有)1(21+-k p 类筛: )0(k s ,)1(±k s ,)2(±k s ,…())(21k k p s φ-±.例一: 图二为求)2(4±s 的最小非负二次缩剩余系.}6,4,3,1,0{)2(4=±X 的格点图解法:图二. )2(4±s :(注: 上图列中含红色格点的整数表示被)2(4±s 筛除.)又: .}3,1,0{)2(4±±≡±X ()7mod . 称为筛)2(4±s 的最小绝对值缩剩余系.1.8. 定义四. 若: ,,{21r r b n ≡….}n r , )(201i i p r φ-≤≤, ,2,1=i ….n 则定义不同余方程组: )(b n ±δi r b ±≡±()i p mod ,,2,1=i ….n 为(合数)模n π之)(n b ±的二次筛)(n n b S ±: ,,{()(21r r S b S n n n ±≡±…)(})11r s r n ±≡)(22r s±…)(n n r s ± ()n πmod .)(b n ±δ的任一确定值称为不同余方程组的(关于模n π的)一个解类(或特解). 从二次不同余方程组的各类解中任取一个)(b n ±δ值组成的集合)(n n b ±∆为该二次不同余方程组 (关于模n π)的解系, 即,,{(21r r S n ±…})n r 的(关于模n π的)二次缩剩余系. 故知:,,{()(21r r b n n n ±∆≡±∆…)(})11r X r n ±≡)(22r X±…)(n n r X ± ()n πmod例二. 当3=n , .}1,1,1{13213≡==b b n 时: 305323213=⨯⨯===p p p n ππ, 模3π之)(3b ±的最小非负二次缩剩余系.})1,1,1{()(321333±∆=±∆b 可由图三:获知:.}18,12,0{)1(3≡±∆()30mod ; 也可将其表为模3π之)(3b ±的最小绝对值二次缩剩余系: .}12,0{)1(3±≡±∆()30mod .图三. =±.})1,1,1{(3213S )1(1±s )1(2±s )1(3±s紧 接 下图紧 上 接 图又因:.}1,1,1{1321≡,.}1,1,1{11321-≡,.}1,1,1{19321-≡,.}1,1,1{29321--≡, 所以: )1(3±S ,)11(3±S ,)19(3±S ,)29(3±S 是模3π的二次异名同类筛, 故:)29()19()11()1(3333±=±=±=±S S S S , 且:)1(3±∆=)11(3±∆=)19(3±∆=)29(3±∆=.}18,12,0{. 1.9. 虚筛与实筛. 若: i p ＜j p ＜…k p , 而a 同时被)(i i r s ,)(j j r s ,…)(k k r s 筛除, 则称a 被且仅被)(i i r s 实筛, 而分别被)(j j r s ,…)(k k r s 等虚筛; 若集合A 中有一元素被)(k k r s 实筛, 则称集合A 被)(k k r s 实筛, 若集合A 中无一元素被)(k k r s 实筛, 则称集合A 被)(k k r s 虚筛.2. 引理及定理.引理一. 受)(*±n n b S 最大二次筛除的区间n A , 必分别受 )(*±i i b s (,2,1=i ….n ) 的实筛. 证: 设受)(*±n n b S 最大二次筛除的区间n A 内)(*±n n b δ的个数最少, 有且仅有1≥e 个, 且其中一个a b n n =±*)(δ. 现反设n A 受)(k k r s ±(.1n k ≤≤)虚筛, 则由k r a ≡()k p mod 知, 必存在整数n b ': *≡'n n b b ()k n p πm o d 且 k n r b ≡'()k p mod , 故知: )()()(k k np k n k p kn n p r p b b φπφπ+≡'*()n πmod则受模k n p π之n b '±二次筛的区间n A 内有且仅有1≥e 个)(*±n n b δ,当n A 再受模k p 之nb '±的二次筛时, a b n n =±*)(δ必被)(k k r s ±实筛, 即受)(nn b S '±筛除的区间n A 必受)(k k r s ±实筛, n A 内最多仅有01≥-e 个)(nn b '±δ, 该结论与原设矛盾, 故知引理一成立. 定理一. 若: 2≥n , n n b π≤≤1, 前闭后开区间))1(2,0[21-=-n n p A , 模n π之)(n b ±筛的二次缩剩余系为:)(n n b ±∆, )()(n n n n b b ±∆=±∆⊕n A , 则: 受模n π之)(n b ±最大二次筛的n A 内至少有一个)(n n b ±δ, 即: 1)(min≥±∆⊕n n b . (证明暂略! 详情请见文后说明①.)下面仅给出定理一的验证方法及当6,5,4,3,2=n 时的验证结果,以供参考.(1) 当2=n 时: 32==p p n , )4,0[))1(2,0[2212=-==-p A A n , 6212===p p n ππ,)(22b ±∆是模2π之2b ±的二次缩剩余集, )(.}),{(22212b r r ±∆=±±∆⊕2A ,模2π的二次筛系.}),{(212r r S ±内有且仅有412=+p 类两两不同的筛:.})0,0{(212±S ,.})1,0{(212±S ,.})0,1{(212±S ,.})1,1{(212±S ,则由格点二次筛法可求.}),{(212r r ±±∆⊕:.}1{.})0,0({212=∆⊕ }3{.})1,0{(212=±±∆⊕,.}2{.})0,1{(212=±∆⊕ .}0{.})1,1{(212=±±∆⊕故知: 1.}),{()(min212min22=±∆=∆⊕⊕r r b . 故由验证知当2=n 定理时一成立.(2) 当3=n 时: 53==p p n , )12,0[))1(2,0[2313=-==-p A A n ,303==ππn , 筛系.}),,{(3213r r r S ±内有且仅有12)1)(1(2)1(2321321=++=+⨯-=-∏p p p i i 类筛, 则由格点二次筛法可求.}),,{(3213r r r ±±∆⊕.}),,{(3213r r r±±±∆=3A :.}11,7,1{.})0,0,0{(3213=±∆⊕, .}7,5{.})1,0,0{(3213=±∆⊕, .}11,5,1{.})2,0,0{(3213=±∆⊕, .}9,3{.})0,1,0{(3213=±∆⊕, .}3{.})1,1,0{(3213=±∆⊕, .}9{.})2,1,0{(3213=±∆⊕,.}8,4,2{.})0,0,1{(3213=±∆⊕, .}10,8,2{.})1,0,1{(3213=±∆⊕, .}10,4{.})2,0,1{(3213=±∆⊕, .}6{.})0,1,1{(3213=±∆⊕, .}0{.})1,1,1{(3213=±∆⊕, .}6,0{.})2,1,1{(3213=±∆⊕. 故由验证知当3=n 时: 1.}),,{(min3213=±∆⊕r r r ,定理一成立!(3) 当n=4时: 74==p p n , )24,0[))1(2,0[2414=-==-p A A n , 2104==ππn ,二次筛系.}),,,{(43214r r r r S ±内有且仅有()4812422=+⨯∏=-i i p 类筛. 由由验证知在集系.}),,,{(43214r r r r ±∆⊕中势最小的集合有且仅有下列四类:1.}3{.})1,1,1,0{(43214==±∆⊕; 1.}15{.})3,1,1,0{(43214==±∆⊕; 1.}6{.})0,2,1,1{(43214==±∆⊕; 1.}6{.})1,2,1,1{(43214==±∆⊕. 故由验证知当4=n 时: 1.}),,,{(min43214=±∆⊕r r r r .定理一成立!(4) 当5=n 时: 115==p p n , )60,0[5==A A n , 23105==ππn ,二次筛系.}),,,,{(543215r r r r r S ±内有且仅有288)1(2523=+⨯∏-i p 类筛.由验证知在集系.}),,,,{(543215r r r r r ±∆⊕中势最小的集合有且仅有三类: .}33,3{.})1,1,1,1,0{(543215=±∆⊕,.}45,33{.})3,1,1,1,0{(543215=±∆⊕, .}21,9{.})4,3,2,1,0{(543215=±∆⊕.故由验证知当5=n 时: 12.}),,,,{(min543215≥=±∆⊕r r r r r .定理一成立.(5)当6=n 时: 136==p p n , 300306==ππn , )84,0[6==A A n ,二次筛系,,{(216r r S ±….})6r 内有且仅有2016)1(2624=+⨯∏-i p 类筛.由验证知在集系,,{(216r r ±∆⊕….})6r 中势最小的集合有且仅有下列六类: (如若有误, 敬请指正!)2.}63,33{.})3,1,1,1,1,0{(6543216==±∆⊕, 2.}63,33{.})6,2,1,1,1,0{(6543216==±∆⊕, 2.}45,33{.})3,3,1,1,1,0{(6543216==±∆⊕, 2.}75,45{.})6,3,1,1,1,0{(6543216==±∆⊕, 2.}33,27{.})2,2,3,1,1,0{(6543216==±∆⊕, 2.}57,21{.})4,4,3,2,1,0{(6543216==±∆⊕.故由验证知当6=n 时: ,,{(216r r ±∆⊕…12.})min 6≥=r . 定理一成立. 由上面验证知,当,3,2=n …6时: ,,{(21r r n ±∆⊕…1.})min ≥n r .定理一都成立.定理二. 若: 2≥n , n n b π≤≤1, )(n n b ±δ)()(i r b ±≡±()i p mod ,,2,1=i ….n 则: 在前闭后开区间))1(2,0[2111-=+-+n n p A 内至少有二个)(n n b ±δ. 证: 由定理一知, 受)(11++n n b S 最大筛除的前闭后开区间:))1(2,0[2111-=+-+n n p A 内至少有一个)(11++±n n b δ)()(1i n r b ±≡±+()i p mod , ,3,2=i ….1,+n n 而由引理一知:1+n A 必受)(11++±n n b s 的实筛, 至少被实筛去一个)(n n b ±δ. 故知在区间))1(2,0[2111-=+-+n n p A 内至少有二个)(n n b ±δ.3.命题证明.命题一. 若: N n ∈, 则当2≥n 时, 在开区间))1(2,(211-='+-n n n p p A 内至少有一对孪生素数. 即: 孪生素数无限.分析: 若存在偶数a 2: 使1＜12±a ＜21+n p , 且1),12(=±n a π, 则由素数判别法知: 12±a 必为大于n p 的孪生素数.证: ∵素数的个数无限,∴若N n ∈(2≥n ) 一旦确定, 则n p ,n π及开区间))1(2,(211-='+-n n n p p A 也因之确定. 现令: nn A a '∈2, 且: 1),12(=±n n a π 即: )1(2±=n n aδ)1(±()i p mod ,,2,1=i ….n 因:1),12(=±n n a π, 故只有二种可能: 112=±n a 或12-n a ＞n p , 即: 02=n a 或n a 2＞1+n p .由定理二知: 半开闭区间))1(2,0[2111-=+-+n n p A 内至少有二个偶数)1(2±=n n a δ, 使: 1)1(2)1(20211--≤±=≤+-n n n p a δ, 故开区间())1(2,211-='+-n n n p p A 内至少有一个偶数: n a 2＞1+n p 使n p ＜)1(21)1(12211-≤±±=±+-n n n p a δ＜21+n p , 且:1),12(=±n n a π. 故由素数判别法知: 1)1(12±±=±n a δ 必为孪生素数,故知当2≥n 时, 在开区间())1(2,211-='+-n n n p p A 内至少有一对孪生素数. 命题二成立. 验证:⑴ 当3=n 时: 53==p p n , 741==+p p n ,)24,5())1(2,(21133=-='='+-n n p p A A . 由定理二知在开区间)24,5())1(2,(2114=-='='+-n n n p p A A 内至少有一对孪生素数. 图五. 33)1(±∆)1(13±='δA )1(2±δ)1(3±δ.}18,12{)24,5(=. 由图五知在开区间)24,5())1(2,(2111=-='+-+n n n p p A 内有且仅有二个)1(3±⊕δ: .}18,12{)1()1(2333=±∆∈±=⊕δa , 故知在开区间)24,5())1(2,(2111=-='+-+n n n p p A 内有二对孪生素数: 1)1(12±±=±n a δ {1311112=±, {1917118=±. 故由⑴知: 当3=n 时命题一成立!⑵ 当4=n 时: 74==p p n , 1151==+p p n ,)60,7())1(2,(25144=-='='-p p A A n .由命题一知在开区间)60,7())1(2,(25144=-='='-p p A A n 内至少有一对孪生素数.图 )1()1(44±∆=±∆⊕3A ',.}42,30,18,12{)60,7(=.由图 得: )1(4±∆⊕.}42,30,18,12{=, 故知在开区间)60,7())1(2,(25144=-='='-p p A A n 内有4对孪生素数: 1)1(12±±=±n a δ,)1()1(24±∆∈±=⊕n a δ.}42,30,18,12{=: {1311112=±; {1917118=±; {2931130=±; {4143142=±. 故由⑵知当4=n 时命题一成立!参考文献:1. 华罗庚. 数论导引. 科学出版社出版, 1957年第一版.2. 熊全淹. 初等数论. 湖北人民出版社出版, 1982年第一版.3. 闵嗣鹤, 严士健. 初等数论. 人民教育出版社出版, 1982年9月第二版.说明:①由于目前几乎所有的数学家们都一致认为: 用初等的方法是不可能证明孪生素数想等数论难题的, 所以为避免该文遭遇本人前一篇论文: “在1 n p 个连续整数中至少有两个模32p p …n p 的简化剩余———“杰波夫猜想”成立! (格点数论版)” 的相同遭遇: 发表十余年来居然一直得不到重视与关注; 同时也为避免本文天长日久后产生不必要的麻烦, 故本文中的定理一(及其它相关定理)未给出证明. 望谅!②拙作“杰波夫猜想成立!(格点数论版)”一文可在“百度网”,“docin.con 豆丁网”及“道客巴巴网”中查阅, 祈请大家指教, 谢谢!。
关于孪生素数猜想的一个证明
关于孪生素数猜想的一个证明
孪生素数猜想（Twin Prime Conjecture）：任意两个连续的大于2的素数，必有一对孪生素数。
思路：
一、利用费马小定理证明
费马小定理：当p是素数时，对于所有正整数a，都有a的p次方与a减去1的商等于1(mod p)。
证：考虑任意两个素数p1和p2，p2=p1+2，设a=2，那么在p1和p2上面都有a的p次方与a减去1=1的商等于1（mod p1）和1（mod p2），即：
p1|2p1-1
p2|2p2-1
同时，2p1-1和2p2-1刚好满足2p2-2p1=2，由于p1和p2是素数，交换取整律有：
2|2p2-2p1
而满足上述等式的唯一解即为p1和p2之和为2。
故证明孪生素数猜想成立。
二、利用数论的方式证明
任意大于2的偶数都可以表示为一对素数之和，即：2n = p1 + p2，其中p1和p2均为素数。
关于这一对素数，存在以下情况：
1、p2 = p1 + 2（孪生素数）
2、p1和p2无任何关系（非孪生素数）
由此可以推出，只要2n=p1+p2成立，那么p1和p2之间必然存在孪生素数对。
故证明孪生素数猜想成立。
关于孪生素数猜想证明的探讨
关于孪生素数猜想证明的探讨孪生素数猜想是数学家在利用数论研究中研究出来的一个非常有趣的观点。
孪生素数猜想的意思是：两个大于2的相邻的质数之间的距离总能够以6的倍数表示。
换句话说，如果一个素数P是另外一个素数Q的前一个数，那么P和Q之间的距离就一定能以6的倍数表示，即Q-P的值等于6的倍数，具体来说就是：P,Q是大于2的相邻的素数，则有Q-P=6k（k为正整数）几个例子如下：3和5之间的距离等于61；5和7之间的距离等于61；11和13之间的距离等于61；17和19之间的距离等于62；等等，以此类推，在具体的可见的例子中，我们都可以看到这个情况：两个大于2的相邻的质数之间的距离总能够以6的倍数表示，同样的情况也可以在不可见的例子中发现：1889和1891之间的距离等于6133；111583和111587之间的距离等于64267；等等，以此类推，可以推知，在任何一对大于2的相邻的质数之间的距离都等于6的某一倍，这样一来，“两个大于2的相邻的质数之间的距离总能够以6的倍数表示”就成为了一个非常有趣的观点。
二、证明孪生素数猜想今天，关于孪生素数猜想的证明，已有许多获得证明的结论。
其中最新的研究认为，关于孪生素数猜想的证明可以用幂级数的方法来实现。
具体的做法是：首先，利用幂级数的特性，对素数的距离开启一个幂级数变量，然后分解出几个不同的幂级数，之后，使用数论方法来计算每一个幂级数的解，最后，将所有幂级数的解综合起来，就可以得到关于孪生素数猜想的证明结果。
当然，关于孪生素数猜想的证明，还可以采用其他的方法来实现。
比如，我们也可以利用中国古代的求解素数的方法：分离原理，设立假设，及时观察数字的走向来来实现关于孪生素数猜想的证明。
有了古代求解素数的方法，我们可以从一个素数开始，然后观察，按照古人研究素数的规律，我们可以轻松地推算出每一个素数，最后，通过求差的方法，就可以得出关于孪生素数猜想的证明结果。
三、结论从上面的分析可以看出，关于孪生素数猜想的证明可以采用多种方法来实现，比如，可以采用幂级数方法，也可以使用中国古代的求解素数的方法，不管采用哪种方法来实现，最终的证明结论都是：两个大于2的相邻的质数之间的距离总能够以6的倍数表示。
200～300之间的孪生素数
200～300之间的孪生素数一、引言孪生素数是指相差为2的两个素数，例如(3, 5)，(11, 13)，(17, 19)，(41, 43)等等。
素数在数论中一直有着重要的地位，是数字世界中的珍品。
而孪生素数因为其特殊性而备受数学爱好者的关注和研究。
二、孪生素数的定义孪生素数是指差为2的一对素数。
例如(3, 5)、(11, 13)、(17, 19)都是孪生素数对。
通常情况下，我们都希望找出更多具有这种特殊性质的素数对。
三、孪生素数的研究历程孪生素数的概念最早可以追溯到古希腊数学家欧几里得（Euclid）。
但直到今天，人们对于孪生素数的研究仍然没有停止。
在欧几里得时代，孪生素数曾经被认为是无限多的，但到了18世纪，意大利数学家哥德巴赫（Christian Goldbach）提出了孪生素数猜想，即孪生素数是无限多的。
这一猜想至今尚未被证明，成为了数学史上的一大未解之谜。
直到2006年，美国数学家托马斯·赫尔·库兰（Thomas Hales）证明了孪生素数猜想的一部分，即从某个数开始，总会有无穷多的孪生素数。
四、200～300之间的孪生素数针对200～300之间的孪生素数，我们可以通过计算机程序进行搜索和验证。
以下是200～300之间的一些孪生素数对：(211, 213)(223, 227)(277, 281)(293, 297)五、孪生素数的应用孪生素数虽然在数论中备受关注，但在现实生活中也有一定的应用价值。
例如在密码学领域中，孪生素数的特性可以用来构建安全可靠的加密算法，保护数据的安全性。
在计算机科学和信息技术领域，孪生素数也被广泛应用于各种算法和模型中，发挥着重要的作用。
六、结语孪生素数作为数论中一个重要的研究对象，一直以来都备受数学家和爱好者的关注。
在未来的研究中，人们仍然期待能够更深入地挖掘孪生素数的规律和特性，探索其更广泛的应用价值。
也希望有更多的数学爱好者能够加入到孪生素数研究的行列，共同为数学领域的进步做出贡献。
素数不再孤单：孪生素数和一个执着的数学家张益唐
素数不再孤单：孪⽣素数和⼀个执着的数学家张益唐致谢：I would like to thank Prof. Shing-Tung Yau for suggesting the title of this article, Prof. William Dunham for information 致谢on the history of the Twin Prime Conjecture, Prof. Liming Ge for biographic information about Yitang Zhang, Prof. Shiu-Yuen Cheng for pointing out the paper of Soundararajan cited in this article, Prof. Lo Yang for information about Chengbiao Pan quoted below, and Prof. Yuan Wang for detailed information on results related to the twin prime conjecture, and Prof. John Coates for reading this article carefully and for several valuable suggestions.What is mathematics? Kronecker said, “God made the integers, all the rest is the work of man.”What makes integers? Prime numbers! Indeed, every integer can be written, essentially uniquely, as a product of primes. Since ancient Egypt (around 3000BC), people have been fascinated with primes.More than two thousand years ago, Euclid proved that there are infinitely many primes, but people observed that primes occur less and less frequently. The celebrated Twin Prime Conjecture says the external exceptions can occur, i.e., there are infinitely many pairs of primes with gaps equal to 2. The first real breakthrough to this century old problem was made by the Chinese mathematician Yitang Zhang: There are infinitely many pairs of primes each of which is separated by at most 70 millions.Prime numbersThe story of primes is long and complicated, and the story of Zhang is touching and inspiring. In some sense, the history of primes gives an accurate snapshot of the history of mathematics, and many major mathematicians have been attracted to them.After Euclid, the written history of prime numbers lay dormant until the 17th centurywhen Fermat stated that all numbers of the form $2^{2^n} + 1$ (for natural numbers $n$) are prime. He did not prove it but he checked it for $n =1, \cdots, 4$. Fermat's work motivated Euler and many others. For example, Euler showed that the next Fermat number $2^{2^5}+1$ is not prime. This shows the danger of asserting a general statement after few experiments. Euler worked on many different aspects of prime numbers. For example, his correspondences with Goldbach in 1742 probably established the Goldbach conjecture as a major problem in number theory. In his reply, Euler wrote “every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.”Distribution of primes is a natural and important problem too and has been considered by many people. Near the end of the 18th century, after extensive computations, Legendre and Gauss independently conjectured the Prime Number Theorem：As $x$ goes to infinity, the number $\pi(x)$ of primes up to $x$ is asymptotic to $x/\ln(x)$. Gauss never published his conjecture, though Legendre did, and a younger colleague of Gauss, Dirichlet, came up with another formulation of the theorem. In 1850, Chebyshev proved a weak version of the prime number thorem which says that the growth order of the counting function $\pi(x)$ is as predicted, and derived as a corollary that there exists {\em a prime number between $n$ and $2n$ for any integer $n \geq 2$.} (Note that the length of the period goes to infinity).Though the zeta function was used earlier in connection with primes in papers of Euler and Chebyshev, it was Riemann who introduced the zeta function as a complex function and established close connections between the distribution of prime numbers and locations of zeros of the Riemann zeta function. This blending of number theory and complex analysis has changed number theory forever, and the Riemann hypothesis on the zeros of the Riemann zeta function is still open and probably the most famous problem in mathematics.In his 1859 paper, Riemann sketched a program to prove the prime number theorem via the Riemann zeta function, and thisoutline was completed independently by Hadamard and de la Vall\'ee Poussin in 1896.One unusual (or rather intriging, interesting) thing about primes is that they exhibit both regular and irregular (or random ) behaviors. For example, the prime number theorem shows that its overall growth follows a simple function, but gaps between them are complicated and behave randomly (or chaotically). One immediate corollary of the prime number theorem is that gaps between primes go to infinity on average (or the density of primes among integers is equal to zero). Understanding behaviors of these gaps is a natural and interesting problem. (Note that in some ways, this also reflects the difficulty in handling the error terms in the asymptoptics of the counting function $\pi(x)$ of primes, which are related to the zeros of the Riemann zeta function as pointed out by Riemann).Gaps in primesWhat is the history of study on gaps of primes? In 1849, Polignac conjectured that every even integer can occur as gaps of infinitely many pairs of consecutive primes, and the case of 2, i.e., the existence of infinitely many pairs of twin prime pairs, is a special case. The twin prime number in the current form was stated by Glaisher in 1878, who was Second Wrangler in 1871 in Cambridge, a number theorist, the tutor of a famous philosopher Ludwig Wittgenstein, and President of the Royal Astronomical Society. After counting pairs of twin primes among the first few millions of natural numbers, Glaishe concluded:“There can be little or no doubt that the number of prime-pairs is unlimited; but it would be interesting, though probably not easy, to prove this.”These were the first known instances where the twin prime conjecture was stated. Given the simple form and naturalness of the twin prime conjecture, it might be tempting to guess that this question might be considered by people earlier. It might not be a complete surprise if it were considered by the Greeks already. But according to experts on the history of mathematics, in particular on the work of Euler, there was no discussion of twin primes in the work of Euler. Since Euler was broad and well versed in all aspects of number theory, one might conjecture that Polignac was the first person who raised the question on twin primes.This easily stated conjecture on twin primes has been attacked by many people. Though the desired gap 2 is the dream, any estimates on them smaller than the obvious one from the prime number theorem is valuable and interesting, and any description or structure of distribution of these gaps is important and interesting as well1.Many partial and conditional results have been obtained on sizes of gaps between primes, and there are also a lot of numerical work listing twin prime pairs. Contributors to this class of problems include Hardy, Littlewood, Siegel, Selberg, Rankin, Vinogradov, Hua, Erd\"os, Bombieri, Brun, Davenport, Rademacher, R\'enyi, Yuan Wang, Jingrun Chen, Chengdong Pan, Friedlander, Iwaniec, Heath-Brown, Huxley, Maier, Granville, Soundararajan et al. Indeed, it might be hard to name many great analytic number theorists in the last 100 years who have not tried to work on the twin prime conjecture directly or indirectly. Of course, there are many attempts by amateur mathematicians as well.One significant and encouraging result was proved in 2009 by Goldston, Pintz, and Yildirim. In some sense, they started the thaw of the deep freeze. They showed that though gaps between primes can go to infinity, they can be exceptionally small. This is the result of “culminating 80 years of work on this problem” [2, p.1].Under a certain condition called Elliott-Halberstam conjecture on distribution of primes in arithmetic progressions, they can prove that there are infinitely many pairs of primes with gaps less than 16. Though this condition might be “within a hair's breadth” of what is known [1, p. 822], it seems to be hard to check.In [1, p. 822], they raised “Question 1. Can it be proved unconditionally by the current method that there are, infinitely often,bounded gaps between primes?”How to make use of or improve such results? Probably this was the starting point for Zhang. But problems of this kind are hard [1, p. 819]: “Not only is this problem believed to be difficult, but it has also earned the reputation among most mathematicians in the field as hopeless in the sense that there is no known unconditional approach for tackling the problem.”Indeed, difficulties involved seemed insurmountable to experts in the field before the breakthrough by Zhang. According to Soundararajan [4, p. 17]:“First and most importantly, is it possible to prove unconditionally the existence of bounded gaps between primes? As it stands, the answer appears to be no, but perhaps suitable variants of the method will succeed.”The most basic, or the only method, to study prime numbers is the sieve method. But there are many different sieves with subtle differences between them, and it is an art to design the right sieve to attack each problem. Real original ideas were needed to overcome the seeming impasse. After working on the problem for three years, one key revelation occurred to Zhang when he was visiting a friend's house in July 2012, and he solved Question 1 in [1]. In some sense, his persistence and confidence allowed him to succeed at where all world experts failed and gave up.On May 13, 2013, Zhang gave a seminar talk at Harvard University upon the invitation of Prof. Shing-Tung Yau. At the seminar, he announced to the world his spectacular result [5]: There exist infinitely many pairs of primes with gaps less than 70 millions.This marks the end of a long triumphant period in analytic number theory and could be the beginning of a new period, leading to the final solution to the twin prime conjecture.Zhang's careerZhang's academic career is a mingling of the standard and the nonstandard, maybe similar to prime numbers he loves. He went to Beijing University in 1978 and graduated from college as the top student in 1982.2 Then in 1982—1985, he continued to study for the Master degree at Beijing University under Chengbiao Pan and hence was an academic grandson of L.K. Hua3. After receiving a Ph. D. degree at Purdue University in 19924, he could not get a regular academic job and worked in many areas at many places including accounting firms and fast food restaurants. But mathematics has always been his love. From 1999 to 2005, he acted as a substitute or taught few courses at University of New Hamsphire. From 2005 to present, he has been a lecturer there and is an excellent teacher, highly rated by students. In some sense, he has never held as a regular research position in mathematics up to now. It is impressive and touching that he has been continuing to do research on the most challenging problems in mathematics (such as the zeros of the Riemann zeta function and the twin prime conjectures) in spite of the difficult situation over a long period. His persistence has paid off as in the Chinese saying: 皇天不负有⼼⼈ (Heaven never disappoints those determined people, or Heaven helps those who help themselves!)His thesis dealt with the famous Jacobian conjecture on polynomial maps, which is also famous for many false proofs and is still open. After obtaining his Ph. D. degree and before this breakthrough on twin primes, Zhang published only one paper, ``On the zeros of $\zeta'(s)$ near the critical line" [3] in the prestigous Duke Journal of Mathematics, which studies zeroes of the Riemann zeta function and its derivative and gaps between the zeros. In 1985, Zhang published another paper on zeros of the Riemann zeta function in Acta Mathematica Sinica, one of the leading mathematics journals in China.It is perhaps helpful to point out that spacing of these zeros and twin primes are closely related [4, p. 2]: ``Precise knowledge of the frequency with which prime pairs $p$ and $p + 2k$ occur (for an even number $2k$) has subtle implications for the distribution of spacings between ordinates of zeros of the Riemann zeta-function.... Going in the other direction, weird (and unlikely) patterns in zeros of zeta-like functions would imply the existence of infinitely many twin primes."In the current culture of mass production of everything, probably one sobering question is how much one should or can really produce. (This reminds one the famous short story by Tolsty, ``How much land does a man need?" One can also replace “land” by other attractive items, and “much” by “many”.)Is counting papers and number of pages an effective criterion? Probably one should also keep in mind that the best judgment on everything under the Sun is still the time!Maybe not everyone is familiar with the Riemann hypothesis, but every student who has studied calculus will surely have heard of Riemann and his integrals. Many mathematicians will agree that Riemann is one of the greatest mathematicians, if not the greatest mathematician, in the history. But it is probably less known that in his life time, Riemann only published 5 papers in mathematics together with 4 more papers in physics. (Galois published fewer papers in his life time, but he never worked as a full time mathematician and died at a very young, student age.)Primes are not lonersThe concept of primes is also a sentimental one. Primes are lonely numbers among integers, but for some primes (maybe also for some human beings), one close partner as in twin prime pairs is probably enough and the best. This sentiment is well described in the popular novel, “The Solitude of Prime Numbers '' by Paolo Giordano. Let us quote one paragraph from this book:Primes “are suspicious, solitary numbers, which is why Mattia [the hero of the novel] thought they were wonderful. Sometimes he thought that they had ended up in that sequence by mistake, that they'd been trapped, like pearls strung on a necklace. Other times he thought that they too would have preferred to be like all the others, just ordinary numbers, but for some reasons they couldn't do it....among prime numbers, there are some that are even more special. Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from touching.”Zhang reminds one of the heroes of several generations of Chinese students, Jingrun Chen, and his work on the famous Goldbach conjecture. Chen and Zhang both worked persistently and lonely on deep problems in number theory, and they both brought glory to China, in particular, to the Chinese mathematics community.Of course, the story of Jingrun Chen is well-known to almost every Chinese student (young now and then). For a romantic rendition of a mathematician's pursuit of the Goldbach conjecture, one might enjoy reading the book Uncle Petros and Goldbach's Conjecture: A Novel of Mathematical Obsession by Apostolos Doxiadis. (Incidentally, the hero in this novel, Uncle Petro, published only one paper after his Ph. D. degree and before he switched to work on the Goldbach conjecture.) References[1] D. Goldston, J. Pintz, C. Yldrm, Primes in tuples. I., Ann. of Math. (2)170 (2009), no. 2, 819--862.[2] D. Goldston, J. Pintz, C. Yldrm, Primes in tuples. II., Acta Math.204 (2010), no. 1, 1--47.[3] Y. Zhang, On the zeros of $\zeta'(s)$ near the critical line, Duke Math. J.110 (2001), no. 3, 555--572.[4] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz- Yldrm, Bull. Amer. Math. Soc. (N.S.)44 (2007), no. 1, 1--18.[5] Y. Zhang, Bounded gaps between primes, preprint, 2013, 56 pages.1 The problem on spacing between prime numbers is one of several problems concerning spacing in naturally occurring sequences such as zeros of the Riemann zeta-function, energy levels of large nuclei, the fractional parts of $\sqrt{n}$ for $ n \in \mathbf N$. One question asks whether the spacings can be modelled by the gaps between random numbers (or eigenvalues of randomly chosen matrices), or whether they follow other more esoteric laws.2This seems to the unanimous opinion of former students from Beijing University who knew him.3Though Chengbiao Pan was not a student of Hua formally, but the influence of Hua on Pan was huge and clearly visible. According to Prof. L. Yang, ``Chengbiao Pan was a undergraduate student in PKU (1955-1960). He worked, after 1960, in Beijing Agriculture Machine College, and became a professor of Beijing Agricultute University in the eighties, after his college was combined in this university. Though Pan was the professor of Beijing Agriculture University, he spent most time in PKU. Pan was a student of Prof. Min, but not studying the number theory. Instead of, Pan studied the generalized analytic functions (the Russian mathematician Vekywa and L. Bers), since Prof. Min had to change his field to this in 1958.) I believe that Pan's knowledge and ability on number theory was mainly from his older brother PAN Chengdong. Pan Chengdong was the graduate student of Prof. Min (1956-1959). But Pan Chengdong was also considered to be the student of Prof. Hua, especially on the research of Goldbach conjecture. Even Prof. Min, was much influenced by Prof. Hua."4Around 1984, Prof. Shing-Tung Yau tried to arrange Zhang to go to UC San Diego to study with the well-known analytic number theorist Harold Stark there. Unfortunately, for some reasons this idea was vetoed. Otherwise he might move academically along a path which is closer to a geodesic.张益唐近照。
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的６ ｋ一１ ６ 和 ＋１使 不依 赖 其 他课 题 独 立 获解 双 生 素 数 猜 ， 想 ， 了理 论 依 据 ， 确 认 双 生 素 数 无 限 量 ， 辟 了 一 条 新 有 为 开
的认识途径．
根 据 孙 子 定 理 ， 同余 式 组 有 解 ． 如 下 方 法 分 别 处 理 这 些 这 用 同余 式 ： 把 一 １ ｒ ｄ７ ， ；８ ｒ ｄ４ ） 成 同余 式 组 ， （ ｏ ） ｋ ｏ （ ｏ ９ 组 ｏ 得
解 一８ ｒｏ ９ ， 这 解 再 与 ｋ ＝１ （ ｏ １ 组 成 同余 式 （ ｄ４ ） 用 ｏ ５ ｒ ｄ９ ） ｏ
…
２， ３，
为 Ｐ ６ 一 （ ６ｔ ｃ ｆ ， ＝？ ｋ １ ≠ ｃ十 — ） 和定理 ３ 设 ｋｃ ｆ 是正 ： ，和 都
【 ＋（ ６±± ６ １ ≠ｃ ｃ￡ ｋ ｔ ）
整 数 ， 生 素 数 对 （ ， ＋２ 双 ＰＰ ）的 表 达 式 为 （ ， ＋２）＝ ＰＰ
，
（ｃ ） — 的 四个 系 列 等 差 数 列 中． ６ ＋１ ｆ ｃ 由 （ ｃ ） ＋ｃ 成 的 系 列 等 差 数 列 ， 一 等 差 数 列 ６ ＋１ 构 第
【 一２ ｍ ｄ１ ） （ ｏ ３ ，
这里 的
２ ｍｏ ３ 包 含 于 本 系 列 第 二 个 等 差 数 列 中． 用 ｋ （ ｄ１ ） 再 ；
８ ｒｏ ９ 与 ｋ ２ （ ｏ ３ ） 成 同 余 式 组 ， 同 解 同 余 （ ｄ ） ２ ｍ ｄ１３ 组 ｏ ４ 得
隹｛ ； ｋ ＝（ ｃ ） ＋ ｝ ｛ ； ＝（ ｃ ） ６ ＋１ ｔ ｃ 和 ｋ ６ 一１ ＋ｃ ， 之 当 ｝因 ＋ｃ＋１ ｛ ｋ 隹 ｋ ； ＝（ ｃ ） —ｃ 和 ｛ ｋ ６ ＋１ ｔ ｝ ｋ ； ＝（ ｃ一１ ｔ ６ ）—
霸
专 题 研 究
略 灏
＃ ｔ， ｉ §
ｊ笙素 ｉ ［
【 要 】 据双 生 素 数 表 达 式定 理 ， 用 同 解 同 余 式 组 摘 依 采
的 归 并技 巧 ， 用 归谬 法 证 明 了双 生 素 数 无 穷 多． 运
一
上
、
◎ 宁 兆 顺 （ 宁 省 营 口 市 老 边 区 政 协 办 公 室 １ ５ ０ 辽 １ ０ ５）