On the Structure of Primes Simon Mark Horvat

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1 On the Structure of Primes Simon Mark Horvat This paper explains the structure of Prime numbers and shows that they are not random. Discussion On pages 322 through 324 of Prime Obssesion, [Derbyshire, 2004], Derbyshire discusses some of the ways in which Primes appear to be random, including the Cramér model which states that...aside from this one restraint on their average frequency, the primes are utterly random... In this paper I show that the Primes are not random and form a lattice-like structure over the Integers. Terms not explained in this paper are documented in a more detailed, unpublished, paper [Horvat 2005] from which this paper has been condensed. Section 1: Structure of Primes over Z Definition: Prime Number Let P Z. We say that P is Prime if its only (+)ve divisors are 1 and P The first 5 negative, and the first 32 positive, Primes in standard representation -7, -5, -3, -2, -1, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127 The sample of Primes extended and expressed in a Modulo 6 grid representation. 3(2n-1) 2(3n-1) 6n-1 6n 6n+1 2(3n+1) 3(2n+1) -(5x5) -(2x3)x (2x3)x3-17 -(3x5) -(2x7) -13 -(2x3)x2-11 -(2x5) -(3x3) -(2x4) -7 -(2x3) -5 -(2x2) x2 5 2x3 7 2x4 3x3 2x5 11 2x3x2 13 2x7 3x5 17 2x3x x3x4 5x5 29 2x3x5 31 5x7 2x3x x3x x3x8 7x7 53 2x3x9 5x x3x x13 2x3x x3x x11 2x3x x3x14 5x x3x15 7x13 5x19 2x3x x3x x3x x3x19 5x23 7x17 2x3x20 11x11 5x5x5 2x3x Author: 2005 Simon Mark Horvat, vsn Page 1

2 173 2x3x29 5x5x7 13x29 2x3x x3x64 5x7x x3x65 17x23 Table 1-1: Integers modulo 6 There is a Lattice-like structure of Composite strands over Z with Prime numbers at the base of each strand. This structure shows more clearly if we leave out the Composite strands of 2n and 3n, n Z, and restrict ourselves to composite strands over the set {6n ± 1}, where the strands are z=x.p i, for all Primes P i >3 and all integers x in {6n ± 1} Fundamental Theorem of Primes: All Integers are members of only one of the sets: Prime Base, B = {±P 1, ±P 2 } = {-3, -2, 2, 3} Subtractive Primes, P S, of the form 2.3.n - 1, n Z Additive Primes, P A, of the form 2.3.n + 1, n Z Composites of the forms 2.n, 3.n, n Z, n 1, ie Multiples of 2 and/or 3, includes 0 Composites, of the form 2.3.n ± 1, n Z, ie Multiples of Primes other than 2 or 3 Theorem: Additive and Subtractive Prime sets All prime numbers ±P i, other than ±2 and ±3, are of the form ±P i = (2.3.n) ± 1 = n.p 2 # ± 1, n Z. ±P i = n.p 2 # - 1 are termed Subtractive Primes, ±P i = n.p 2 # + 1 are Additive Primes. The set ±P i = (2.3.n) ± 1 is closed under multiplication and division I term the set, {6n ± 1}, the Prime Candidates (PC), P. Figure 1-1 shows stylised representations of the Composite strands over the set {6n ± 1} for the range [-31, 187]. Also, it does not include the Composite strands of 2 and 3. Each horizontal unit is of size 1. Each vertical unit is of size 6 and is referred to as a unit of size K, due to the PC Key function, K(r), defined below. For clarity I have only included strands for n=0, ±1, ±2, ±3, ±4. From left to right the graphs are Composite strands of: {6n -1}, {6n ± 1}, {6n +1}, 5 and 25 {all intersections with {6n±1} of the strand of 25 overlap with the strand of 5} 5 and 7, 11 and and and 25 Just plotting these few strands shows part of why it has been so difficult to find the Structure of Primes over Z ; Primes are indirectly related, and are the result of gaps in the Composite weave of smaller Primes Author: 2005 Simon Mark Horvat, vsn Page 2

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4 Theorem: Prime Candidate Holes x = 6n ± 1 is Prime iff it is not the intersection of a Composite strand of a smaller Prime Candidate, x' = 6m ± 1, with the Prime Candidate Set, n,m Z,0 m n This is really just the Sieve of Eratosthenes restricted to the Prime Candidate set and can be equivalently stated as: Theorem: Prime PC Holes x = 6n ± 1 is Prime iff it is not the product of a smaller Prime Candidate, x' = 6m ± 1, n,m Z,0 m n Let P i ± be the i th Subtractive or Additive Prime Candidate, P i ± = 6i ± 1, P i - = 6i 1, P i + = 6i + 1, i Z. Define the PC Key function K( r ) st 1) K r = r 1,r R, r 0 6 2) K r = r 1,r R 6 Therefore, i N, K(P i ) = i K(P i + ) = i K(P i ± ) = i K( r ) is the distance of r from zero in terms of Prime Candidate Integer Key values Section 2: Prime Density and the Prime Number Theorem The Prime Number Theorem approximates the number of primes that are less than a given number n [Ingham 1932] [Jameson 2003] [Weisstein (2)] Theorem: Prime Number Theorem The number of primes that are less than a given number n is approximately n/ln(n), n n ln n It's clear now why the density of Prime Numbers decreases as n increases; the Prime Candidate Strands (multiples) of P n ± intersect with the Prime Candidate Set and prevent a Prime Candidate from being Prime, resulting in fewer Prime Candidate Holes as z increases. Euclid's Proof though, shows that existing Composite strands do not fill all of the Prime Candidate Set but instead must always leave Prime Holes forming the start of new Composite strands over P While this cause is straightforward, its effect is not. Attempts to calculate the Number of Primes by Combinatorial methods based on this structure are quickly affected by Complexity Theory. This simple structure has quite tangled results Section 3: Factoring Composites Take any integer z. Divide z by 2 until there are no more factors of 2, to obtain y Author: 2005 Simon Mark Horvat, vsn Page 4

5 Divide y by 3 until there are no more factors of 3, to obtain x By the Fundamental Theorem of Primes x is now of the form: i) 6c-1, or ii) 6c+1 for some integer c Subtractive Prime Candidates are of the form 6c-1. IF 6c-1 is not Prime then 6c-1 = (6n-1)(6m+1) has non-trivial Integer solutions for all 3 PC Key values c, n, m. Solving for PC Key value m m (6n-1) = c-n m= c n 6n 1 IF (6c-1) is a multiple of (6n-1) then (c-n) is a multiple of (6n-1) and (c-n) mod (6n-1) 0 c mod (6n-1) n 6c-1 mod (6n-1) 0 6c mod (6n-1) 1 Solving for PC Key value n n (6m+1) = c+m n= c m 6m 1 IF (6c-1) is a multiple of (6m+1) then (c+m) is a multiple of (6m+1) and (c+m) mod (6m+1) 0 c mod (6m+1) -m 6c-1 mod (6m+1) 0 6c mod (6m+1) 1 Additive Prime Candidates are of the form 6c+1. IF 6c+1 is not Prime then either 1) 6c+1 = (6n-1)(6m-1), or, 2) 6c+1 = (6n+1)(6m+1) have non-trivial Integer solutions for all 3 PC Key values c, n, m. In each case we need only solve for one term, n or m, WLOG Case 1) Product of Subtractive Prime Candidates: Solving for PC Key value m m (6n-1) = c+n m= c n 6n 1 IF (6c+1) is a multiple of (6n-1) then (c+n) is a multiple of (6n-1) and (c+n) mod (6n-1) 0 c mod (6n-1) -n 6c-1 mod (6n-1) 0 6c mod (6n-1) 1 Author: 2005 Simon Mark Horvat, vsn Page 5

6 Case 2) Product of Additive Prime Candidates: Solving for PC Key value n n (6m+1) = c-m n= c m 6m 1 IF (6c+1) is a multiple of (6m+1) then (c-m) is a multiple of (6m+1) and (c-m) mod (6m+1) 0 c mod (6m+1) m 6c+1 mod (6m+1) 0 6c mod (6m+1) -1 Note: P n ± and P m ± may be themselves be Composite Prime Candidates for some, but not all, values of n and m. There must be at least one n and m for which these PCs are not Composite, even if it is for PC Key values of 1 and C Initial PC Key value = =1 c 6 In both cases, factoring a Subtractive or an Additive PC, one can safely use this value as the Initial PC Key value Section 4: PC Multiplication Charts PC Multiplication charts demonstrate the relationships between PCs, especially when we look at the charts in terms of their PC Integer Key values The difference between (6n-1)x(6m+1) and (6{n+1}-1)x(6m+1) is m = 6(6m+1) The difference between (6n-1)x(6m+1) and (6n-1)x(6{m+1}+1) is n = 6(6n 1) Table 4-1: Subtractive Row x Additive Column PC Multiplication Chart The difference between (6n-1)x(6m-1) and (6{n+1}-1)x(6m-1) is m = 6(6m 1) The difference between (6n-1)x(6m-1) and (6n-1)x(6{m+1}-1) is n = 6(6n 1) Author: 2005 Simon Mark Horvat, vsn Page 6

7 Table 4-2: Subtractive x Subtractive PC Multiplication Chart The difference between (6n+1)x(6m+1) and (6{n+1}+1)x(6m+1) is m = 6(6m+1) The difference between (6n+1)x(6m+1) and (6n+1)x(6{m+1}+1) is n = 6(6n+1) Table 4-3: Additive x Additive PC Multiplication Chart Table 4-4: Subtractive Row x Additive Column Multiplication Chart in PC Key Values Table 4-5: Subtractive x Subtractive PC Multiplication Chart in PC Key Values Author: 2005 Simon Mark Horvat, vsn Page 7

8 Table 4-6: Additive x Additive PC Multiplication Chart in PC Key Values Columns in Tables 4-4 and 4-5 have magnitude reflected in PC Key Column 0, with sign not reflected Rows in Tables 4-4 and 4-6 have magnitude reflected in PC Key Row 0, with sign not reflected Both Rows and Columns are reflected between Tables 4-5 and 4-6, resulting in a 180 o rotation around the intersection of PC Key Row 0 and Column 0 Section 5: Generalised Prime Lattice Structures Given the simplicity of the Prime structure I wondered what is it about the Prime structure which gives us Primes? On further consideration it is clear that the underlying property of the Prime structure itself is in the term ±1 Definition: The Generalised Prime Lattice Structure B z ={0} {±P i e i,±p j e j,...,±p k e k }, S z =z n 1, A z =z n 1, z, n Z, z 0, If B {z} ={0} then z=0 else z=p i e i.p j e j.... P k e k where B(z) is the Prime Base for this set, and B(z) may be written using the product, ie B(6), or as a set of primes, ie B({±2, ±3}) or B({2, 3}). ±1 is an element of all Generalised Prime Candidate sets, zn ±1. It is the result of the cases with z and/or n equal to zero. The X Base Primes in B z ={0} {±P i e i,±p j e j,...,±p k e k }, if any, are the first X Primes in B(z) in order of size. Note that these Base Primes in B(z) are the set {±P i e i,±p j e j,...,±p k e k }, and not the set {±P i 1,±P j 1,...,±Pk 1 }, unless e i =1 for all i, ie, the Base Primes of B(245)=B({5,49}) are 5 and 49 I now term Primes of the form 6n ±1, the Natural Primes, or B(6) = B(2, 3) = B(±2, ±3), with the set of Base Primes {±2, ±3}. In the Fundamental Theorem of Primes the Prime Base was labelled as B = {±2, ±3}. I will retain B as indicating the special case of the Prime Base of the Natural Primes The set of Prime Candidates under B(z), {zn ±1}, is closed under multiplication Author: 2005 Simon Mark Horvat, vsn Page 8

9 (z n 1) x (z m 1) = z 2 nm zn zm +1 = z (znm n m) +1 (z n 1) x (z m +1) = z 2 nm +zn zm 1 = z (znm +n m) 1 (z n +1) x (z m +1) = z 2 nm +zn +zm +1 = z (znm +n +m) +1 Note these are generalised forms of the multiplicative equations for the Natural Primes, with z in place of 6 (6n 1) x (6m 1) = 6 2 nm 6n 6m +1 = 6 (6nm n m) +1 (6n 1) x (6m +1) = 6 2 nm +6n 6m 1 = 6 (6nm +n m) 1 (6n +1) x (6m +1) = 6 2 nm +6n +6m +1 = 6 (6nm +n +m) +1 {zn ±1} is also closed under division, with sub-factors of form zn ±1 or is itself Prime in B({z}), meaning that it does not decompose into sub-factors of form zn ±1 other than its own (+)ve or ( )ve value and ±1 The PC Key function K(x) is now extended to K Z (x), the value of n in zn ±1 under B(z). I will retain K(x) as an alternative shorthand nomenclature equivalent to K Z (x) in the current Prime Base under consideration With B(z), we are not studying all integers; instead we are investigating the Patterns of the Prime Lattice structures formed over the subsets of Subtractive and Additive Prime Candidates under B(z), {zn ±1}. Theorem 11 is a corollary to Theorem 9, providing a definition for Prime PCs in B(z) Theorem: Prime PCs in B(z) x = zn ± 1 is a Prime PC in B(z) iff it is not the product of a smaller Prime Candidate in B(z), x' = zm ± 1, z,n,m Z,z 0,0 m n ie, in B(10), P 2 - = 19, P 2 + = 21 There are neither Prime, nor Composite, PCs in B(0) as it only has the 2 PCs, ±1 All of B(1)'s Primes are the same as B(6)'s Primes, but in B(1): There are no Primes in the Prime Base, B(1) = {±1} ±2 and ±3 are Prime PCs, not Base Primes All Primes are both Subtractive and Additive PCs y is a Prime PC in B(y -1) and B(y +1) for all Integers y, y>1 B(z) Prime PC is an equivalent term to Prime PC in B(z) For example, not counting 1, the first few (+)ve Prime PCs in B(10) are: 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69, 71, 79, 89, 91, 101, 109, 111, 119, 129,... Many B(10) Prime PCs are not Natural Primes, ie not Prime in B(6): 9, 21, 39, 49, 51, 69, 91, 111, 119, 129,... Many B(10) Prime PCs are not even Prime Candidates in B(6): 9, 21, 39, 51, 69, 111, 129,... The first few Composite Prime Candidates in B(10) are: 81 = 9x9 99 = 9x11 Author: 2005 Simon Mark Horvat, vsn Page 9

10 121 = 11x = 9x = 9x = 11x = 11x = 9x29 10n-1 10n 10n (2x5)x (2x5) x x5x x5x x5x x5x x5x x5x x5x8 9x9 89 2x5x9 91 9x11 2x5x x5x x5x12 11x x5x x5x x5x x5x x5x17 9x x5x x21 2x5x x5x x19 2x5x x5x x5x23 11x x5x89 9x9x11 There are 15 (+)ve Prime PCs in B(10) before the first B(10) Composite PC, 81 { K 10 (81)=8 }, then 2 more B(10) Prime PCs before the next Composite B(10) PC, 99 { K 10 (99)=10 }, totaling 17 B(10) Prime PCs below 99 There are 7 (+)ve Prime PCs in the Natural Primes, B(6) { 5, 7, 11, 13, 17, 19, 23 } before the first Composite PC, 25 { K(25)=4 }, then 2 more Primes before the next Composite PC, 35 { K(35)=6 } Thus, the Weave of the Prime Lattice varies depending on the value of z. Examining the composites formed by set of Prime Candidates under B(z), {zn ±1}: K Z ( (z n 1) x (z m 1) ) = znm n m K Z ( (z n 1) x (z m +1) ) = znm +n m K Z ( (z n +1) x (z m +1) ) = znm +n +m Many of the properties documented in this paper hold for B(z), ie Factorisation Algorithms in B(z), there will be a Prime Number Theorem applicable to each B(z), z>0 Author: 2005 Simon Mark Horvat, vsn Page 10

11 For small values of n, or m, the above multiples are affected more by the value of z. As zn ±1 tends towards infinity the effect of z decreases and, for z>6, the Density of Primes in Prime Candidates under B(z) decreases for the same reason, and in the same manner, as the density of Primes in the Prime Candidates of the Natural Primes, B(6) Theorem: Infinity of Prime PCs in B(z) : There are an infinite number of Prime PCs in B(z), z>0 Lattice Structures over R The domain of z in B(z) may be extended in at least two ways to include the Real numbers. Case 1: Extend the Domain of the z term to R In this extended domain the Real Prime Base, B(r), has an infinite number of members, r n, all of the integer powers of the Real number r B z ={r n }, S z =r n 1, A z =r n 1, n Z,r R,r 0, Case 2: Extend the Domain of n to R Another way to extend the domain into the Reals is to extend n into the Reals instead of z. This subset is formed by taking a Real number, r 0, and all of the Reals an integer displacement from r 0 n={r 0 y },r 0 R, y Z Then B z ={0} {±P i e i,±p j e j,...,±p k e k }, S z =z {r 0 y} 1, A z =z{r 0 y} 1, z Z, z 0, r 0 R, y Z Lattice Structures over C A Gaussian Integer is a Complex number z=a+bi, where a and b are integers. Gaussian Integers can be uniquely factored in terms of Gaussian Primes [Weisstein (3)] A Gaussian Prime is a Gaussian Integer which is only divisible by itself and 1, and by no other Gaussian Integer. [Loy (1)] Definition: Gaussian primes are Gaussian integers satisfying one of the following properties [Weisstein (4)]. 1. If both a and b are nonzero then a+bi is a Gaussian prime iff a 2 +b 2 is a Natural Prime. 2. If a = 0, then bi is a Gaussian prime iff b is a Natural Prime and b 3 (mod 4). 3. If b = 0, then a is a Gaussian prime iff a is a Natural Prime and a 3 (mod 4). The Natural Primes which are also Gaussian Primes are 3, 7, 11, 19, 23, 31, 43,... [Sloane A002145]. 2 is a Natural prime, but it is not a Gaussian prime because it has Gaussian Integer factors, (1 i)(1+i) = = 2 Theorem: A Prime Candidate Structure of Gaussian Primes Author: 2005 Simon Mark Horvat, vsn Page 11

12 For Gaussian Prime z = a+bi. If a = 0, or b = 0, then if z >3 and the Integer Key Value, n=k( z ) is: - Even, then z is a Subtractive Prime, 6n 1, n>1, - Odd, then z is an Additive Prime, 6n +1, n>0, Note that this Theorem only describes the structure of a subset of the Gaussian Primes. There may be more Prime Candidate, or other, types of structures to consider. Section 6: Generalising Euclid's Proof of the Infinitude of Primes [not included in this condensed extract from the unpublished paper [Horvat 2005] ]... Section 7: Symmetric Prime Lattices In this Section, we examine Symmetric Prime Lattices, refer Figure 7-1: "Symmetries of {6n ±1} over [-31, 343]". Figure 7-1 shows that every pattern of Composite strands has a cycle of length 6 x Product of the strands and 180 o Rotational Symmetry repeated at half of this cycle length, starting from 0 K. ie, a pair of Composite strands z i =6n i -1 and z j =6n j +1 have a cycle of length 6n i n j, or n i n j K, and repeated 180 o Rotational Symmetry at every 3n i n j, =(n i n j )/2 K, starting from 0 K. The cycle and symmetry length properties hold regardless of the number of strands involved. In Figure 7-1, from left to right, the graphs are Composite strands of: Strand(s) Cycle Length Points of Symmetry K 6.5n K K 12.5n K 5, 7 35 K (=6x35 =210) 17.5n K (=3x35n =105n) 5,11 55 K 27.5n K 5,13 65 K 32.5n K 7,11 77 K 38.5n K 5,17 85 K 42.5n K 7,13 91 K 45.5n K 7, K 59.5n K 7, K 66.5n K 11, K 71.5n K 5,7, K 192.5n K 17,19,23,25 185,725 K 92,862.5n K Table 7-1: Symmetries of {6n ± 1} n Z These symmetries hold for all B(z), for Integer values of z, with z being the size of unit K. I have not investigated these symmetries yet in terms of Real z Author: 2005 Simon Mark Horvat, vsn Page 12

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14 References: Caldwell, Chris K., Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1? Derbyshire, John, 2003, Prime Obsession. ISBN: Chapter 18 Hoffman, Paul, 1998, The Man Who Loved Only Numbers. ISBN: ) P Erdős was the master of elementary methods... 2) P32-33 Euclid's Proof of the Infinitude of Primes Horvat 2005, The Structure of Primes and Prime Lattice Theory. Unpublished Paper Hui, K, Infinitely many Primes of form 6n+5. Ingham, A. E., 1932, The Distribution of Prime Numbers. ISBN: Jameson, G. J. O., 2003, The Prime Number Theorem. ISBN: Loy, J, (1) Gaussian Primes. Loy, J, (2) Primes and 6. RSA, RSA Challenge. Sloane A Weisstein, Eric W. (1) "Fundamental Theorem of Arithmetic." From MathWorld--A Wolfram Web Resource. Weisstein, Eric W. (2) "Prime Number Theorem." From MathWorld--A Wolfram Web Resource. Weisstein, Eric W. (3) "Gaussian Integer." From MathWorld--A Wolfram Web Resource. Weisstein, Eric W. (4) "Gaussian Prime." From MathWorld--A Wolfram Web Resource. Author: 2005 Simon Mark Horvat, vsn Page 14