View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CSCanada.net: E-Journals (Canadian Academy of Oriental and... ISSN 1923-8444 [Print] Studies in Mathematical Sciences ISSN 1923-8452 [Online] Vol. 6, No. 2, 2013, pp. [96{103] www.cscanada.net DOI: 10.3968/j.sms.1923845220130602.478 www.cscanada.org Related Distribution of Prime Numbers Kaifu SONG[a],* [a] East Lake High School, Yichang, Hubei, China. * Corresponding author. Address: East Lake High School, Yichang 443100, Hubei, China; E-Mail: skf08@ sina.com Received: March 6, 2013/ Accepted: May 10, 2013/ Published: May 31, 2013 Abstract: The surplus model is established, and the distribution of prime numbers are solved by using the surplus model. Key words: Surplus model; Distribution of prime numbers Song, K. (2013). Related Distribution of Prime Numbers. Studies in Mathematical Sci- ences, 6 (2), 96{103. Available from http://www.cscanada.net/index.php/sms/article/ view/j.sms.1923845220130602.478 DOI: 10.3968/j.sms.1923845220130602.478 1. SURPLUS MODEL In the range of 1 ∼ N, r1 represents the quantity of numbers which do not contain 1 a factor 2, r = [N · ] ([x] is the largest integer not greater than x); r represents 1 2 2 2 the quantity of numbers which do not contain factor 3, r = [r · ]; r represents 2 1 3 3 4 the quantity of numbers do not contain factor 5, r = [r · ]; ... 3 2 5 In the range of 1 ∼ N, ri represents the quantity of numbers which do not contain factor pi , the method of the surplus numbers ri−1 to ri is called the residual model of related distribution of prime numbers. In the rangep of 1 ∼ N, excluding all multiples of prime numbers which are not greater than M (M ≤ N), then r is the surplus number. For the factor that rM rM [ ] ≤ r, so there are at least [ ] related primes in the range os M. N N 96 Song, K./Studies in Mathematical Sciences, 6 (2), 2013 2. RELATED DISTRIBUTION OF PRIME NUMBERS 2.1. Distribution of Prime Numbers Exclusion the multiples of 2, 3, 5 in natural numbers, the rest numbers can be categorized as eight numeric axes: A1, B1, A3, B3, A7, B7, A9, B9 (Axis A: 3n + 2, axis B: 3n + 1). On number axis (A-axis or B-axis), r0 represents the quantity of numbers which 8 do not contain factors 2, 3, 5, 7, r = [ ·N]; r represents the quantity of numbers 0 35 1 p1 − 1 which do not contain prime factor P1 (P1 > 7), r1 = [r0 · ]; r2 represents the p1 p2 − 1 quantity of numbers which do not contain prime factor P2, r2 = [r1 · ]; ... p2 pi − 1 ri = [ri−1 · ]. pi The method of the surplus numbers ri−1 to ri is called the surplus model of distribution of prime numbers. 2 Sn is the quantity of prime numbers in the range of N, make M = pn, by 2 pn − 1 M pn − 1 pn Sn = [rn−1 · · ] = [rn−1 · · ]: (1) pn N pn N 2 pn−1 − 1 pn−1 Sn−1 = [[rn−2 · ] ]: (2) pn−1 N We can obtain (pn − 1)pn Sn = [ 2 · Sn−1]; pn−1 8 × 10 × 11 8 p1 − 1 pn−1 − 1 where S1 = [ ] = 25. (rn−1 = [:::[ · N] ]:::] ]). 35 35 p1 pn−1 (pi − 1)pi (pi − 1)pi For the fact that 2 > 1, so f[ 2 · Si−1]g (i = 1; 2; :::; n) are pi−1 pi−1 ascending series in which all numbers are not less than 25. Therefore, lim Sn = 1, n!1 there are infinite prime numbers in natural numbers. 2.2. Distribution of Primes in Arithmetic Series S(a;r) is the quantity of prime numbers which have the form an+r((a; r) = 1), then Sn pn(pn − 1) S(a; r) = = [ 2 · Sn−1]; '(a) '(a)pn−1 where S1 = 25. ('(a)) is Euler function of a.) a is a constant, '(a) is a constant. There are infinite prime numbers which have the form an + r((a; r) = 1). 97 Related Distribution of Prime Numbers 2.3. Distribution of Prime Numbers Between n2 and (n + 1)2 2 2 2 2 0 Make n = pn,(n + 1) = (pn + 1) , then Sn is the quantity of prime numbers in 2 the range of (pn + 1) , " # 0 (pn + 1)(pn + 1 − 1) Sn = 2 · Sn : pn " # 0 (pn + 1)(pn + 1 − 1) [Sn] Let f(n) = Sn − Sn = 2 · Sn − [Sn] = [ ], where S1 = pn pn 25. (pn − 1) (pn−1 − 1) (pn−2 − 1) (p2 − 1) pi − 1 For f(n) = [ [ [ [:::[ 2 · S1:::], and > 1. pn−1 pn−2 pn−3 p1 pi−1 Then lim f(n) = 1. n!1 The quantity of prime numbers between n2 and (n + 1)2 are increasing along with the increasing of n. In a small range, there is at least one prime number between n2 and (n + 1)2. Therefore, there is at least one prime number between n2 and (n + 1)2; and the quantity of the prime number are increasing along with the increasing of n. 2.4. The Interval Between Primes pi−1 and pi During the distribution of prime numbers, pn (pn − 1) (pn−1 − 1) p3 − 1 p2 − 1 Sn = [ [ [ [:::[ [ · S1]:::]: pn−1 pn−2 pn−3 p1 p1 Let pi − pi−1 be the average interval between two primes pi−1 and pi, then in 2 the range of pn, 2 2 pn pn pn−1 p2 p1 pi pi − pi−1 = = [ [ [:::[ [ ]:::]; > 1: Sn pn−1 pn−1 − 1 p2 − 1 S1 pi − 1 and lim pi − pi−1 = 1. n!1 There is no maximum value of the average interval between primes pi−1 and pi. Therefore, the maximum value of the average interval between primes pi−1 and pi tends to infinity. 2.5. Distribution of Twin Primes On number axis pair (A1 − B3 or A7 − B9 or A9 − B1), r0 represents the quantity 1 of numbers which do not contain factors 2, 3, 5, 7, r = [ · N]; r represents the 0 14 1 p1 − 2 quantity of numbers which do not contain factor p1, r1 = [r0 · ]; r2 represents p1 p2 − 2 the quantity of numbers which do not contain factor p2, r2 = [r1 · ];... ri = p2 pi − 2 [ri−1 · ]. pi 98 Song, K./Studies in Mathematical Sciences, 6 (2), 2013 The method of the surplus numbers ri−1 to ri is called the surplus model of distribution of twin primes. 2 Ln is the quantity of twin primes in the range of N, let M = pn, by 2 pn − 2 M pn − 2 pn Ln = [rn−1 · · ] = [rn−1 · · ] (3) pn N pn N 2 pn−1 − 2 pn−1 Ln−1 = [[rn−2 · ] ]: (4) pn−1 N We can obtain that (pn − 2)pn Ln = [ 2 · Ln−1]; pn−1 11 × 9 1 p1 − 2 pn−1 − 2 where L1 = [ ] = 7. (rn−1 = [:::[ N] ]:::] ]). 14 14 p1 pn−1 (pi − 2)pi (pi − 2)pi For the fact that 2 > 1, so f[ 2 · Li−1]g (i = 1; 2; :::; n) are pi−1 pi−1 ascending series in which all numbers are not less than 7. Therefore, lim Ln = 1, n!1 there are infinite twin primes in natural numbers. 2.6. Distribution of Twin Prime Numbers Within Ten There are two pairs of twin prime numbers within 10, these two pairs of twin prime numbers are called twin prime numbers within ten. For example, 11, 13, 17, 19 are twin prime numbers within ten. On number axis pair (A1 −B3 −A7 −B9), r0 represents the quantity of numbers 1 which do not contain factors 2, 3, 5, 7, r = · N; r represents the quantity of 0 70 1 p1 − 4 numbers which do not contain factor p1, r1 = r0 · ; r2 represents the quantity p1 p2 − 4 pi − 4 of numbers which do not contain factor p2, r2 = r1 · ;... ri = ri−1 · . p2 pi The method of the surplus numbers ri−1 to ri is called the surplus model of distribution of twin prime numbers within 10. 2 Tn indicates the quantity of twin prime numbers within 10, let M = pn, by 2 pn − 4 M pn − 4 pn Tn = rn−1 · · = rn−1 · · : (5) pn N pn N 2 pn−1 − 4 pn−1 Tn−1 = rn−2 · · : (6) pn−1 N We can obtain that (pn − 4)pn Tn = 2 · Tn−1; pn−1 99 Related Distribution of Prime Numbers 11 × 7 ∼ 1 p1 − 4 pn−1 − 4 where T1 = = 1:10. (rn−1 = · N · · ::: · ). By 70 70 p1 pn−1 pn(pn − 4) pn−1(pn−1 − 4) pn−2(pn−2 − 4) p2(p2 − 4) Tn = 2 · 2 · 2 · ::: · 2 · T1 pn−1 pn−2 pn−3 p1 pn (pn − 4) (pn−1 − 4) p3 − 4 p2 − 4 = · · · ::: · · · T1 pn−1 pn−2 pn−3 p1 p1 p3 − 4 p2 − 4 pi − 4 For the fact that · ·T1 > 1, > 1 (i = 4; :::; n), so lim Tn = 1, p1 p1 pi−2 n!1 there are infinite twin prime numbers within 10 (For example, 36 pairs in the range of 105). 2.7. Distribution of n2 + 1 Prime Numbers Lemma 7.1 4(mp+10x)2 +1 ≡ 4(mp−10x)2 +1 (modp), where p is an odd prime. Proof. 4(mp + 10x)2 + 1 − 4(mp − 10x)2 − 1 = 16mxp: Lemma 7.2 If x ± y 6≡ 0 (mod p), then 4(mp + 10x)2 + 1 6≡ 4(mp + 10y)2 + 1 (mod p). Proof.