A curious relation between

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dimension expansion, by modular arithmetic, of a prime sequence that we ... 1.1 Arithmetic congruence is a modular relation between a pair of integers, and a ...
Sophie Germain Primes

T.H. Ray June 2012

A curious relation between Sophie Germain primes & Modulo 12 arithmetic. T. H. Ray* Abstract As the atoms of arithmetic, prime integers have important and generally understood applications to the one-dimension line of natural integers, N*. Here we introduce a twodimension expansion, by modular arithmetic, of a prime sequence that we conjecture is unique for being closed under modular continuation; i.e., there are no higher prime pairs, mod 12, beyond our identified sequence where the initial point is 0(mod12) and the result graphs to a perfect 12 hour clock.

*Thomas H. Ray 1209 Norwood Rd Lansing, MI 48917 [email protected]

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Sophie Germain Primes

T.H. Ray June 2012

1. 0 Introduction 1.1 Arithmetic congruence is a modular relation between a pair of integers, and a closed sequence of congruent integers is a one-dimension sequence in a bounded two-dimension field. 1.2 The 3-dimension kissing number (n=3) of Euclidean spheres is known to be 12, proved in the late 19th century. The 4-dimension kissing number was proved to be 24, by Musin in 2003. Of n > 4 , the numbers are known to be 240 for n=8 and for n=24, 196,560. One can find a history for research supporting these results on the MathWorld web site.1 All the known values are congruent, modulo 12. 1.3 Consider the table:

S n Kissing Order k in relation to low dimension Geometry & Topology 1 Order* 1 2 3 4

!(0) = "3 !(1) = "2 !(2) = "1

!(3) = 0 5 !(4) = 1

2

3

4

5

Geometry

Topology

k

N

Point Line Plane Sphere

None

0 2 6 12

0 1

Hypersphere

0

S S1 S2 S3

24

!3(!21 ) 3(2 2 ) 3(2 3 )

1.4 [Ray, 2006]2 demonstrates that a well ordering of the natural numbers N can be expressed as a function of the self organization of the universal set of complex numbers, Z that does not require the axiom of choice. 1.5 We encounter the two dimensional complex plane (row 3, column 1) at the exact 1 topology ( S column 3) and value (-1) that one expects of a line made of eternally recurring parity pairs of elements self similarly extended on ! , because the point is normalized by multiplying the hyperbolic geometry of the point by the hyperbolic geometry of the line. The exponent, 1, then (row 3, column 5), is the least counting element of a discretely defined order of n-spheres (corresponding to e, the root of the natural logarithm ln). Odd-even parity is preserved as a continuous function of the evolving n-sphere radius defined by the order of kissing spheres. 1.6 Rotations on ! incorporate all the geometry up to the hypersphere inclusive, where 1 3 time is a simple parameter of reversible trajectory between S and S . Four dimension analysis is algebraically closed in a two dimension complex model.

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Sophie Germain Primes

T.H. Ray June 2012

2.0 Method 2.1 We will demonstrate natural well ordering by attacking an open problem in geometry and topology—the n-dimension kissing number problem, in which the unit 2-sphere recurs as the three-dimension manifold of least order, 3(2 3 ) = ! n =1 = 24 in the ndimension order of Euclidean sphere kissing numbers, n > 2 . We take the initial condition as the least congruent (mod 12) Sophie Germain pair (11,23), corresponding to the S1 kissing number (12). S1 is the zeroth member of the Euclidean sphere kissing sequence. 2 n continues the 2-dimensional complex plane analysis over n dimension manifolds. 2.2 Suppose we construct a congruence subgroup of Sophie Germain primes, mod 12, compact and closed under summation of dimension order terms. We conjecture that maximum density by least action would include the subspace 0 + 1 as a recurring 2sphere (S 2 ) independent of the n-dimension group. We mean, in a non-lattice packing, every dimension group, n ! 3 , contains at least one 12-vertex lattice (simplex) as the zeroth member sphere. We guarantee congruence (mod 12) by iterative application of the Sophie Germain property between parallel rows of integers forming cells of 4 nodes, of which at least 2 nodes are Sophie Germain* and separated by the least order of the desired multiples. Thus: 11*

23*

23*

47 is the initial cell. Although 47 is not Sophie Germain, it connects the next

least Sophie Germain prime by the least multiple of 3(2 n ) , which happens to be n = 6, or 192: 11*

23*

23*

47

239*

23*

479*

11* (0)

(1)

491* (0)

(1)

23*

We sum 47 + 192 to derive 239. Then:

(4)

47

tom ray 8/1/08 10:31 AM Formatted: Numbered + Level: 1 + Numbering Style: 1, 2, 3, ... + Start at: 1 + Alignment: Left + Aligned at: 0.21" + Tab after: 1.13" + Indent at: 1.13", Tabs:Not at 0.5"

239* (2)

tom ray 8/1/08 10:31 AM Deleted:

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Sophie Germain Primes

T.H. Ray June 2012

The prime 479 is only coincidentally Sophie Germain; as we saw from the previous iteration, it need not be necessarily so (though all integers from our generating function will be either Sophie Germain primes, or “safe” primes in which p of 2p + 1 is Sophie Germain. In the figure, we have begun to number the iterations (n). The smaller-font integer, (n), associated with the forward arrows, represents the order of 3(2 n ) , n > 1, which implies binomial doubling with each +1 ordinal increase (Table 2):

tom ray 8/1/08 11:59 AM Deleted:

Table 2. Linear order correspondence to exponential growth.

!N =

ORDER

N= 3(2 n ) , n =

N=

3 4 5 6

24 48 96 192 ...

1 2 3 4 2.3 The complete sequence is

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11, 23, 239, 479, 491, 683, 1559, 1583, 3359, 7103, 7151 derived from: (1) (9) 11*

(2) 23*

(0) (1)

23*

(3) 479*

(4) 491*

(0)

(5) 683*

(3119)

3167

(983)

(8) 3359*

(14207) (14303)

(4) (4)

239*

(7)

(4)

(4)

47

(6)

1367

(1)

1559*

(5)

1583*

6719

(2)

7103*

7151*

*Sophie Germain (k) unconnected nodes (i) iteration

2.4 When we ignore the discrete numerical results, the global pattern resembles a continuous function with four apparently discontinuous nodes: (0) (1)

(0)

(4)

(4)

(4) (4)

4

(1)

(5)

(2)

Sophie Germain Primes

T.H. Ray June 2012

2.5 We divide the cells discretely, with redundant arrows, to derive:

tom ray 8/1/08 12:01 PM Deleted:

(1)

(2)

(6)

(7)

(3)

(4)

(8)

(9)

(5)

!N 9 (Table 1) appear to be random and therefore meaningless, the sum ! 24 is ! N =1 . We detect 5 unique patterns (1—4, 6). Note that while orders of magnitude

i= 2

This will be important later.

2.6 When we reconnect the unique patterns in order to avoid redundancy, we encounter discontinuity:

(1)

(2)

(3)

(4)

(6)

We want to compactify the object so as to maintain continuity, eliminate redundant nodes and eliminate unconnected nodes. 2.7 The arithmetic congruence of the elements (mod 12) joins endpoints at iterations (1,9) to eliminate the redundant 23. Reversing and superimposing the terminating cell on the first, produces: 11* 23*

7151*

47 (1,9) 5

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Sophie Germain Primes

T.H. Ray June 2012

We find that we can compactify (4,5) by the same rule, eliminating unconnected nodes: 491*

683*

1539*

1367 (4,5)

So that compactification leaves exactly one prime integer at each node and the sequence is complete:

11*

23*

479*

491*

(0)

(0)

(1,9)

(4)

239* (2)

3167

(4)

(4)

7151* 47

683*

(4) (+1)

1559* 1367 (3)

(4,5)

3359*

(5)

1583* (6)

6719 (7)

(+2)

7103* (8)

Figure 1. The Sophie Germain Maniifold

By eliminating the redundant 23, the sum is ordered over the entire sequence of iterations 9

to ! N =1 : ! 24 i=0

(fig 1)

2.8 The manifold possesses the same Euler Characteristic, ! = +1 (15-16+2, by the Euler graph formula of vertices – edges + faces) as characterizes the real projective plane. 3 The manifold is therefore compact and nonorientable. The generating function resembles a “clock” (fig. 2) that by eliminating the redundant 23:

6

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Sophie Germain Primes

T.H. Ray June 2012

7151

0 (23)

11

7103

3359

23

IX

III

1583

239

479

491 1559

VI 683

Figure 2. Sophie Germain Clock, the underlying 2manifold disc of Sophie Germain orbifold.

3.0 Result Complete & closed?

!

3.1 We conjecture that the sequence is complete and closed; i.e., no higher N exists to continue from either the terminating 7151 or the Sophie Germain prime (14303) generated by it. (The final prime is only coincidentally Sophie Germain and not part of our sequence.) The conjecture is bold. However, there are heuristic and theoretical arguments to suggest that the sequence is self limiting to this exact range in the domain of prime integers.

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Sophie Germain Primes

T.H. Ray June 2012

3.2 The heuristic argument is based on the fact that Sophie Germain strings are self limiting in principle. Because sufficiently large primes are restricted to terminating digits 1,3,7,9—and because integer 1 is eliminated in the first iteration of the Sophie Germain property (2p+1), integers 7 and 3 in iterations following (these result in terminating digits of 5 and 7)—the case of 9-ending prime integers remains as the possible initial condition for an infinite 1-dimensional string of Sophie Germain primes. The likelihood diminishes quickly (by the prime number theorem) for this fast growing function. Then: The likelihood of continuation of our 2-dimensional sequence of connected Sophie Germain primes is similarly self-limiting. The terminating digits 1 and 3 (7151, 14303) allow ending digits 3 and 9 possibilities to admit primes with the Sophie Germain

!

property to the next least N ; however, even though the probability for continuation doubles by moving up to 2 dimensions, we doubt that the fast growing function can overcome the thinning of primes by the prime number theorem. 3.3 The second reason, theoretic, that we think our sequence of Sophie Germain primes is complete and closed, is: The approach to unity quotient of terminating members (7103/7151) is the closest convergence for the connected sequence, and, perhaps for any pair of Sophie Germain primes separated by N Order. Significantly, as we specify n > 1, and the terms (7103/7151) are separated by Order 2, or 3(2 4 ) , as indicated in Table S1, we call the pair “Twin Sophie Germain Primes.” In our topological context,

! N = 0 ( 3(22 ) implies unit

2-sphere. Separation by ! N = 2 , 3(2 4 ) , therefore implies twins. In other words, the quotient pair (7103/7151) converges to unity in 2 dimensions, as closely as a 1dimensional prime number ratio, P /P + 2 for arbitrarily large P (twin primes). Are there more twin Sophie Germain primes, with slowly converging ratio, like twin primes, or do they end with this complete and self limiting sequence?—the sequence of twin primes is demonstrably not self limiting. We conjecture that there is only one pair of twin Sophie Germain primes. Therefore, the n-dimension manifold is compact.

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Sophie Germain Primes

! 2 (48)

T.H. Ray June 2012

7151

0 (23)

11

!0

7103

23

IX

III

1583

479

!0

!1 (24)

491 1559

VI

Figure 3. Correspondence between congruence and closure in the Sophie Germain field. Figure 3 shows that from the initial condition to the terminating pair, there is symmetry among the least congruent pairs (12, or ! 0 ), the next order (24, !1 ) and the “twin” Sophie Germain pair (48, ! 2 ).

1 Weisstein, Eric W. "Kissing Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KissingNumber.html

2

Ray, T. “Self organization in real and complex analysis,” Proceedings of the sixth international conference on complex systems, New England Complex Systems Institute. 2006. 3

Thurston, W.P. [1980] “The Geometry & Topology of 3-Manifolds,” electronic notes distributed by Princeton University via web, published at http://www.msri.org/publications/books/gt3m.

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