自然数八度周期律暨丢番图四定理_英文_

自然数八度周期律暨丢番图四定理_英文_ The Octet Periodic La w of Natural Numbers and Ξ the Four Theorems of Diophantus S HAO Meng ( )Changchun Tractors Manufacturing Plant ,Changchun 130031 ,China Abstract : On the basis of discovering the Octet Periodic Law of natural numbers , the four theorems of Diophantus are formulated. This solves exhaustively the old problem of decomposing any positive integer into a sum of squares. Key words : periodic table ; Octet Periodic Law ; ocete operator ; embryonic number ; discriminant ; discriminator ; four theorems of Diophantus () CL C number : O15611 Document code : A Article ID : 052926579 20040420023205 mat hematical properties of t he positive i ntegers a re 1 The Octer Periodic La w of Natural periodic f u nctions of t heir octet codes . Numbers 2 The Manipulation of Octet Codes We denote the set of natural numbers {1 , 2 , 3 , } by N. Thus , 0 | N. While we are travelling along the chain of τ Consider the finite set of the octet codes { A , B , successors by starting from 1 , we feel inwardly that the τC , D , E , F , G , H} ?. Using trivial logical operation , it properties of natural numbers pulsate at a pace of eight is easy to work out the algorithmic expression consecutive integers. Thus , drawing inspiration from the C ?G = E Periodic Law of Chemical Elements , we discovered the i1e1 , the product of an element of group C by an element Octet Periodic Law of Natural Numbers. Each such an of group G transforms into an element of group E. eight consecutive integers forms a recurring period , the In the same way , we can have B ?D = H , A ?C ( elements of which belong to the group s 8 n - k k = 7 ,6 , = C , E ?E = A , etc . ) ,1 ,0, which are designated as the group s A , B , C , Fig11 gives the mathematical logic system of the D , E , F , G , and H respectively. The symbols A , B , multiplication operation of the octet codes by letting x ?y , H are called specifically octet codes. be the entry in row x and column y . Arranging the elements of N in this way , it is Multipliers recognized that , there is a periodic tendency towards A B C D E F G H recurrence of similar properties , thereby producing the ABCDEFGHA octet periodic table of natural numbers , as shown in table BDFHBDFHB CFADGBEH1 . In the table , the prime numbers are shown in C DHDHDHDHD boldface. All the representative elements in any given EBGDAFCHE vertical column in the table have the same representation FDBHFDBHF () as a sum of the least numberof squares. Elements in GFEDCBAHG the same row of the table are said to be in the same H H H H H H H H H period. Thus this building- up scheme brings to natural Fig11 Multiplication of octet codes numbers the most naturel principles of classification and the most natural hierarchy of method. The fundamental We see from above that the octet codes A , B , , idea expressed by these classifications comes to be known H possess both the properties of numbers and operators , as the Octet Periodic L aw of Nat ural n umbers : The so it is convenient to call them octet operators among their affinity tending to combine with the operators they which we are particularly interested in the odd operators “hardness”of C , acted on. Furthermore , owing to theA , C , E , and G which are responsible to the 2 2 G , we do transform neither CG into E , nor C, G, representation of any positive integer as a sum of squares. 2 2 2 2 or C, CG, Ginto A . By convention , we call C , G , the “ hard ” Using Fig11 , it is readily to evaluate the powers of operators because of their eccentric and egotistic the octet operators and the results are listed in Table 2 .characters , and A , E the “soft ”operators because of Ta b11 The Octet Periodic Ta ble of Natural Numbers ?8 n - 4 E ?8 n - 3 F ?8 n - 2 G ?8 n - 1 H ?8 n D A ?8 n - 7 B ?8 n - 6 C ?8 n - 5n ?????????????????? 1 1 2 3 4 5 6 7 8 2 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 3 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 5 6 41 42 43 44 45 46 47 48 7 49 50 51 52 53 54 55 56 8 57 58 59 60 61 62 63 64 9 65 66 67 68 69 70 71 72 10 73 74 75 76 77 78 79 80 11 81 82 83 84 85 86 87 88 12 89 90 91 92 93 94 95 96 13 97 98 99 100 101 102 103 104 14 105 106 107 108 109 110 111 112 15 113 114 115 116 117 118 119 120 16 121 122 123 124 125 126 127 128 17 129 130 131 132 133 134 135 136 18 137 138 139 140 141 142 143 144 19 145 146 147 148 149 150 151 152 20 153 154 155 156 157 158 159 160 21 161 162 163 164 165 166 167 168 22 169 170 171 172 173 174 175 176 23 177 178 179 180 181 182 183 184 24 185 186 187 188 189 190 191 192 25 193 194 195 196 197 198 199 200 ? Perf sq. ; ?Duad sqs. ; ? Trio sqs. ; ?Quad sqs. Ta b12 The po wers of the Operators A B C D E F G H 2 2 r 2 2 r 2 2 r 2 B = D = C= A = D = GCEFGr r + 1 r A = A D= HH= H 2 r + 1 r + 2 r + 2 2 r + 1 2 r + 1 E= EF= HB = HC= CG= G Note : r ?N 311 Octet Discriminant 3 Problems of the Representation of Every natural number n , n > 1 , can be uniquely Any Positive Integer a s a Sum of 2expressed in the form a a a1 2tSquaresn = pp p t 12 α?N , each pis a prime , and p?pfor i ?j . where i i i j The problems of decomposing integers of various If we cross out pairwise the identical prime factors , types into the sums of squares are one of the main topics the resulting expression would be of study in mumber theory which have not been tackled 1 n = rr rwith s ? t , or n ? øsystematically as yet. By introducing the new idea of1 2s ris some member of { p, p, , p} , and n where periodicity of natural numbers and restating the past works i 1 2 t is called the embryonic number of n , and the empty øof Fermat , Euler , and Lagrange , we are in a position to solve the old problem exhaustively. set . () () If we replace each of the prime factors in the n - 5 ?3 mod 838 expression of n by the octet operator pertaining to it and Hence 2 2 group together the like operators , then we have a () x + y , 8 - 5 mod 8; d , by using transient expression which can be simplifie1e1 , we cannot decompose any integer of the form i Fig11 and Table 2 , to the form 8 n - 5 into a sum of two squares. γγ1k Θ Θ =qqor ? ø 1k Whether an integer of the form 8 n - 5 can beγwhere = 1 or 2 , k = 1 ,2 ,or 3 , and each qis an expressed as a sum of three squares ? i i k = 8 n - 5 is the sum of three Suppose that Θ octet operator . is called the discriminant because it 2 2 2 () gives the nature of the mumber of representations. squares. Then k ?3 mod 8= x+ y+ zfor some This procedure is best illustrated by the example of integers x , y and z . Θ finding the discriminant of 396805500 . It is obvious that among the three integers , either two integers are even and one integer is odd , or all of the Decomposing completely the given integer into prime () factors , we have three are odd. Thus , from 1, we have 3 3 22 2 2 396805500 = 3?13 ?7 ?19 ?5?17 ?2 x+ y+ z? and () 5 mod 8,for 2 8 x ,2 | y ,4 8 y ,4 | z ; n = 3 ?13 ?7 ?19 ?5 ?17 () 1 mod 8,for 2 8 x ,2 | y ,4 8 y ,2 | z , ()4 Hence 4 8 z ; or 2 8 x ,4 | y ,4 | z ; Θ = C ?E ?G ?C ?E ?A = () 3 mod 8,for 2 8 x ,2 8 y ,2 8 z . 2 2 2 2 A ?C?E?G = A ?C?A ?G ?CG () From the above congruences and 3, we see that Θ The various types of the discriminant are called 2 22() ?8 n - 5 mod 8, x + y + z the discriminators , amounting to 21 members , i1e1 , A , 8 x ,2 8 y and 2 8 z for 2 2 2 2 E , 2 , 2 A , 2 E , C , C, G , G, CG , CG, i1e1 , any integer of the form 8 n - 5 is decomposable into 2 2 2 2 2 2 CG , CG, 2 C , 2 C, 2 G , 2 G, 2 CG , 2 CG, a sum of three squares of odd integers. 2 2 2 2 CG , 2 CG。Θ Corollary 1 Any integer with discriminant = C 312 Behaviour of the Elements in Group C 2 or CGcan be expressed as a sum of three squares. For the integrity of our scheme , we have to restate 313 Behaviour of the Elements in Group F the works of some eminent mathematicians in the simplest Proposition 1 No integer of the form 8 n - 2 is the linguistic fashions in the sequel . sum of two squares , but it can be expressed as a sum of Proposition 1 No integer of the form 8 n - 5 is the three squares. sum of two squares , but it can be expressed as a sum of Proof Suppose that k = 8 n - 2 is the sum of two 2 three squares.22() for some squares . Then k ?6 mod 8and k = xy+ Proof Using the language of congruences , integers x and y . suppose that k = 8 n - 5 is a sum of two squares. Then k 2 2 n - 2 is even , the Because integer of the form 8 () ?3 mod 8and k = x+ yfor some integers x and integers x and y must be both even or both odd. Thus , y . 22() x+ y?0 ,2 orfrom 1and the relevant congruence Because integer of the form 8 n - 5 is an odd () x and y , we see that it is 4 mod 8for the specified number , we may let one integer , say x , be odd , then 22( ) congruent to 6 mod 8 . x+ yimpossible to make the other integer y be even. n - Hence we cannot decompose any integer of the form 8 For a positive integer u > 1 , we have 2 into a sum of two squares. () 1 mod 8,for 2 8 u ;2 Now , suppose that k ?8 n - 2 can be expressed as () ()u ? 4 mod 8,for 2 | u ,4 8 u ; 122() k ?6 mod 8= x+ ya sum of three squares. Then () 0 mod 8,for 4 | u . 2 z fom some integers x , y , and z . + Thus Because integer of the form 8 n - 2 is even , the () 1 mod 8,for 2 8 x ,4 | y ,22 ()2x+ y? x , y and z must be either two odd and one even integers () 5 mod 8,for 2 8 x ,2 | y ,4 8 y . ( ) or all three even. Thus , from 1 and the relevant But 2 2 2 () congruence x+ y+ z?0 ,2 ,4 or 6 mod 8for the n - 3 appear successively as then the integers of the form 4 specified x , y anelements in group s A and E , repeatedly. Consequently d z , we see that . 2 22 () x + y+ z?8 n - 2 mod 8,group B includes all the elements of group s A and E for 2 8 x ,2 8 y ,2 | z ,4 8 z multiplied by 2 . Θ i1e1 , any integer of the form 8 n - 2 can be expressed as = 2 ,2 A , or 2 E Proposition 1 Any integer with 2 can be expressed as a sum of two squares. a sum of three squares , one of which is even and is not 2 2 Θ divisible by 16 , and the other two are odd. Proposition 2 Any integer with = 2 C,2 G, 2 2 2 Θ 2 CG,or 2 CG can be expressed as a sum of three Corollary 1 Any integer with = 2 C ,2 G ,2 CG 2 squares. or 2 CGcan be written as a sum of three squares. This property is also self- evident by judging from the 314 Behaviour of the Elements in Group G distribution of representatons of the sums of squares in - 1 is the Proposition1 No integer of the from 8 n sum of two or three squares , but it can be expressed as a group B of the Octet Periodic table of natural numbers. 2 317 Beha iour of the Elements in Group D sum of four squares. The elements in group D are of the form 8 n - 4 = () Proof Since 8 n - 1 is odd , we see from 2and 2 2 () (( ) 4that , the sum of two squares , say k = x+ y,or 4 2 n - 1. If n = 4 m - 3 ,4 m - 2 ,4 m - 1 ,or 4 m m 2 2 2 ) = 1 ,2 ,3 , , then the integers of the form 2 n - 1 the sum of three squares ,say k = x+ y+ z,for appear successively as elements in group s A , C , E , some x , y and z , and 8 n - 1 are incongruent modulo 8 . Hence we cannot decompose an integer of the form 8 n - 1 and G , repeatedly. Consequently , group D includes all into a sum of two or three squares. But we can express it the elements of A , C , E , and G multiplied by 4 . as a sum of four squares , since the sum of four squares , Thus , the rules of discriminant applied to group s A , C , 2 2 2 2 (say k = x+ y+ z+ w, is congruent to 8 n - 1 mod E , and G hold equally well for group D . ) 8for 2 8 x ,2 8 y ,2 8 z ,2 | w ,4 8 w . 318 Behaviour of the Elements in Group H 2 3 Θ Corollary 1 Any integer with = G or CG can The elements in group H are of the form 8 n = 2n . be written as a sum of four squares. Since n can be any integer from 1 upward , the prime- 315 Behaviours of the Elements in Groups A and E power decomposition of n can be any combination of the Θ Proposition 1 Any integer with = A or E can primes. Thus , the rules of discriminant used for the be expressed as a sum of two squares. foregoing group s hold universally for group H. 2 r- 1 Euler proved Fermatπs assertion that every prime of () Moreover , if n = 28 t - 1for r , t ?N , the the form 4 m + 1 is decomposable uniquely into a sum of integer of group H is representable as a sum of four 1 ,2 two squares. Further , we know that if two integers squares. It is evident that as n increases , the additional 2 are sums of two squares , then so is their product. If integers which are expressible as sums of four squares m > 1 is even in 4 m + 1 , then the prime is of the form become scarcer . ) ( 8 n - 7 group A ; if m ?1 is odd , then it is of the319 Summary ) ( form 8 n - 3 group E. Hence the above property is Summing up the foregoing results , we give below 2only an extension of Fermatπs assertion. Note that in the the Four Theorems of Diophantus general cases , the decomposition into a sum of two Diophantine Theorem 1 An integer can be squares is not necessarily unique . Θ expressed as a sum of two squares if its discriminant is 2 2 Θ Proposition 2 Any integer with = C, G, A , E , 2 , 2 A or 2 E. 2 2 CG,or CG can be expressed as a sum of three squares. Diophantine Theorem 2 An integer can be J udging from the distribution of representations of the Θ expressed as a sum of four squares if its discriminant is 2 2 2 2 2 () sums of squares in group A for C, G, or CG, G or CG. () or in group E for CG, of the Octet Periodic table of Diophantine Theorem 3 An integer can be natural numbers , this property is self- evident . Θ expressed as a sum of three squares if its discriminant 2 2 2 2 2 2 316 Behaviour of the Elements in Group B is C , C, G, CG , CG, CG, 2 C , 2 C, 2 G , 2 2 2 2 2 The elements in group B are of the form 8 n - 6 = 2 G, 2 CG , 2 CG, 2 CG or 2 CG. Another way of(( ) ) 2 4 n - 3. If n = 2 m - 1 or 2 m m = 1 ,2 ,3 , , Θ saying the same thing is that if the nonempty is such that number can be decomposed into a sum of two squares , 2 Θ | { A , E ,2 ,2 A ,2 E} ?{ G , CG} i1e12 2then the integer can be expressed as a sum of three + 4080 987654321 = 31161squares. Exa mple 2 Express the integer 999999996 as a Diophantine Theorem 4 For the perfect squares , sum of squares. the following three properties are true by intuition. The prime-power decomposition of 999999996 is 2 ( 999999996 = 2?3 ?83333333Property 1 An integer of the form 8 n - 7 group ) A is a perfect square iff Thus n = 3?83333333 2 Θ ( ) n = k- k + 2Π2 with k ?Nand = C E = G. ( By Diophantine Theorem 2 , the given integer can be Property 2 An integer of the form 8 n - 4 group ) expressed as a sum of four squares , such as Dis a perfect square iff 2 2 2 2 2 2 () n = 2 k- 2 k + 1 with k ?N999999996 = 215811+ 110+ 13+ 3= 2 2 2 2 ()Property 3 An integer of the form 8 n group H 31622+ 220+ 26+ 6 is a perfect square iff Exa mple 3 Express the integer 89708234 as a sum 2 of squares. n = 2 kwith k ?N In other words , an integer is a perfect square iff its The factorization of 89708234 into primes is Θ ?ø.89708234 = 2 ?7 ?11 ?19 ?23 ?31 ?43 = n Thus This gives an example that elementary mathematics 3 3 Θ can be used to solve some old problems in number theory = 2 ?G ?C ?C ?G ?G ?C = 2 CG= 2 CG. the given integer is once a highly original notion comes into our minds. By Diophantine Theorem 3 , expressible as a sum of three squares , such as The validity of the Four Theorems of Diophantus is 2 2 2best illustrated by the following examples : 89708234 = 9469+ 192+ 97 Exa mple 1 Find out the representation as a sum of References : squares for 987654321 , the ladder number of nine 1 MORRIS , KL INE. Mathematical thought from ancient to terraces. modern times M . New York : Oxford University Press , Decomposing completely the given integer into prime 1972 : 608 - 612. factors , we have 2 UNDERWOOD DIDL EY. Elementary number theory M . 2 2 987654321 = 3?17?379721 2nd edit . San Francisco : W. H. Reeman and Company , Θ Thus n = 379721 and = A 1978 : 10 - 19 , 20 , 27 - 31 , 141 - 148 , 149 - 154. According to Diophantine Theorem 1 , this ladder 自然数八度周期律暨丢番图四定理 邵 檬 ()长春拖拉机制造厂 , 吉林 长春 130031 摘 要 : 在发现自然数八度周期律的基础上 , 明确而系统地制定了丢番图四定理 , 从而准确 、彻底 、全面地解决了数论 中任意正整数究竟能分解成几个平方数之和的这一古老问题 。 关键词 : 周期表 ; 八度周期律 ; 八度算子 ; 胚胎数 ; 判别式 ; 判符 ; 丢番图四定理 中图分类号 : O15611