“My general view of mathematics is that most of the complicated things we learn have their origins in very simple
examples and phenomena.” – Dr. Richard Evan Schwartz, Chancellor's Professor of Mathematics, Brown University
"Seek simplicity and distrust it." – Alfred North Whitehead
Introduction
This site was created to introduce a deterministic algorithm
and geometry, in the form of a spiral sieve, that reveals the 8-dimensional progression
which ultimately determines the distribution of prime numbers. Hidden deep within this
geometry, a variant form of modulo 30 wheel factorization, are revealed
mysteriously beautiful symmetries, profound in their implications. Although labeled a "sieve," it's more than that. It not only serves
as an efficient prime number sieving algorithm that accounts for all (as opposed to "almost all") prime numbers
≥ 7, but it also demonstrates how prime numbers are distributed within a radial geometry that effectively defragments the
Ulam Spiral and ultimately leads us to "the theory of everything." This domain, once fathomed,
reveals itself to be a beautiful mathematical object in and of itself.
The math required to understand what follows, elementary arithmetic and geometry, has been mastered by most high school students.
For the sake of universal accessibility, this is for the most part a narrative approach to the subject, analogous to "writing through
the curriculum." Testimony supporting this approach comes from Stephen Hawking in his book, Brief History of Time: A Reader's Companion
(1992), where he states: "Equations are necessary if you're doing accountancy, but they are the boring part of mathematics. Most of the
interesting ideas can be conveyed by words or pictures." What follows is a mixture of well known fact and conjecture (the latter so labeled)
informed by more than two decades of heuristic experimentation complemented by research. In any event, if your intent is to better understand
how the prime number sequence is constructed, read on. In this you will not be disappointed.
If this site "proves" anything, it's that those with only an elementary education in mathematics can have direct access to the
beautiful symmetries encompassing algorithmic order embedded within the seeming chaos of the prime number distribution and related patterns.
It's hard to imagine a presumably enlightened "Spirit of the Universe" barring the vast majority of intelligent beings who haven't pursued higher
mathematics a measure of insight into the profound mystery "curled up" in the roots of the number universe. After all, the
experience is exhilarating, if not spiritualizing.
Since its inception this site has attracted more than 6,100 "absolute unique visitors" from 136 countries (many of whom have returned repeatedly)
and not one has offered counter-examples or otherwise refuted what follows. The proof is in the pattern. Res ipsa loquitur ...
Foundations
The genesis of most if not all repeating prime number patterns described in the mathematics literature (Twin Primes, Cullen Primes,
Chen Primes, Sexy Primes, Cousin Primes, Sophie Germain Primes, Siamese Primes ... the list goes on and on ...) can be readily deciphered
using this sieve as an analytical tool employing modular arithmetic
(and specifically, mod 30 relationships). The most obvious of these patterns are the three "Twin Prime Distribution Channels" described
at some length, below. These and all other such repeating–albeit intermittent and seemingly
random–patterns are fundamentally sub-patterns of the set of natural numbers not divisible by 2, 3 and 5 when arrayed in 8 dimensions,
whether in a matrix or spiral form, as discussed in great detail below. Yes, this is a bold assertion, but once you've apprehended the beautiful
cascading symmetries and ever-repeating and expanding chordal progressions described in detail herein, you'll come to the same inescapable
conclusion. Here are two examples supporting this claim, i.e. using this sieve to analyze and predict Siamese Primes
(n2 -2 and n2 +2 are primes) and Sophie Germain Primes (p and 2p+1 are primes).
The foundation and key to the array populating the sieve are the 8 integers dubbed "prime roots."
These 8 foundational numbers (
1, 7, 11, 13, 17, 19, 23 and 29) are arrayed in the first inner rotation of the sieve (or, alternatively, the first row of an 8-column matrix).
These are the first 8 integers of the infinite set of all natural numbers not divisible by 2, 3 and 5 (for convenience, labeled PRS for "Prime Root Set")
which by definition includes (and only includes) 1 and all primes ≥ 7 and their multiples (and, as you'll see below, it's conjectured
that the entire set can be generated by a simple expression involving 2, 3 and 5). [Note: A mathematician would describe this set as:
primes greater than 5 are of the form 30n + r, where r is one of {1, 7, 11, 13, 17, 19, 23, 29}, i.e., in base 30 all primes greater than 5 must end
in 1, 3, 7 or 9. Also note that if you plug the number 30 into Euler's totient function,
phi(n): phi(30)= 8, with the 8 integers (known as totatives)
smaller than and having no factors in common with 30 being: 1, 7, 11, 13, 17, 19, 23 and 29, i.e., what are called
"prime roots" above. Thirty is the largest integer with this property.]
The integer 30, the product of the first three primes (2, 3 and 5), and thus a primorial, plays a powerful role organizing the array's
perfect symmetry, viz., in the case of the 8 prime roots:
1+29=30; 7+23=30; 11+19=30; and 13+17=30.
In The Number Mysteries well-known mathematician Marcus Du Sautoy writes: "In the world of mathematics, the numbers 2, 3, and 5
are like hydrogen, helium, and lithium. That's what makes them the most important numbers in mathematics." Although 2, 3 and 5 are the only prime numbers
not included in the domain under discussion, they are nonetheless integral to it: First of all, they sieve out roughly 3/4ths of all natural numbers,
leaving only those nominally necessary to construct a geometry within which the primes can be optimally arrayed. The remaining 26.6666666...%
(to be a bit more precise) constituting the array can be constructed with an elegantly simple interchangeable expression that incorporates the
first three primes. It's conjectured that this expression generates ALL (and only) the numbers in the array (and their negatives):
xnyn ± zn where x=2, y=3 and z=5.
Thus: xnzn ± yn and ynzn ± xn.
Given that all primes >5 are in the array, it is conjectured that all primes >5 can be generated by this expression.
Deep Symmetries
The intervals between the prime roots (and every subsequent row or rotation of the sieve) are perfectly balanced, with a
period 8 difference sequence of: {6, 4, 2, 4, 2, 4, 6, 2}. The entire domain can thus be defined
as 1 {+6 +4 +2 +4 +2 +4 +6 +2} {repeat ... ∞}. These intervals total 30, and each rotation is separated by 30, as
well. The first two rotations, therefore, increment as follows:
Interestingly, the sum of the 2nd rotation = 360. Is it coincidental that the product of the first three primorials,
2, 6 and 30 = 360? Or is it coincidental that when you multiply the first five
Fibonacci numbers in sequence, you produce 1, 2, 6 and 30? And, speaking of the Fibonacci number sequence, there is symmetry mirroring
the above in the relationship between the terminating digits of Fibonacci numbers and their index numbers equating to members of the array
populating the Croft Spiral Sieve:
Remarkably, the sequence of Fibonacci terminating digits indexed to the prime roots,
13,937,179 (see graphic, above),
is a prime number and a member of a prime pair (with 13,937,177), though, if you're curious, not a reversible prime
(although the reversal is a semi-prime: 9,461 * 10,271 = 97,173,931, and you'll note that both its factors have two combinations summing to 10).
In addition, 13,937,179 when added to its reversal 97,173,931 = 111,111,110 and the entire repeating (and palindromic) Fibo sequence end-to-end
(equivalent to two rotations around the sieve) gives you this equivalency: 1,393,717,997,173,931 ≡ 11,111,111 (mod 111,111,110)...
[And interestingly, 11,111,111 * 111,111,110 = 1234567876543210 and 111,111,110/11,111,111 = 10.]. Also 1,393,717,997,173,931 is divisible by the
repunits 11 and 1,111 and 11,111,111. [Author's note: It's astounding
that not only do these beautiful Fibonacci terminating digit patterns correlate by index number to the first 16 elements in the set of all natural
numbers not divisible by 2, 3 and 5 but that the 2nd set of 8 index numbers, equivalent to the 2nd rotation around the sieve, sums to 360.]
Echoing the Fibonacci patterns just described, the terminating digits of the prime roots (17,137,939), when added to their
reversal (93,973,171) = 111,111,110. And, when you connect the prime root terminating digit sequence to its reversal,
the entire palindromic sequence end-to-end produces this: 1,713,793,993,973,171 ≡ 111,111,111 (mod 111,111,110)
[And in this case, 111,111,111 * 111,111,110 = 12345678876543210.]. And if that isn't enough, 1,713,793,993,973,171 is also divisible
by the repunits 11 and 1,111 and 11,111,111.
Well, not quite enough, because there's yet another related dimension of symmetry: The terminating digits of the prime root angles
(24,264,868; see illustration of Croft Spiral Sieve, below) when added to their reversal (86,846,242) = 111,111,110, not to
mention this sequence possesses symmetries that dovetail perfectly with the prime root and Fibo sequences, including the fact that when
it is connected to its reversal (giving us 2,426,486,886,846,242), it's divisible by the repunits 11 and 1,111 and 11,111,111.
And when you combine the terminating digit symmetries described above, capturing three rotations around the sieve in their actual
sequences, you produce the ultimate combinatorial symmetry:
Here's yet another fascinating dimension of symmetry: the pattern of 9's created by decomposing and summing either the digits of
Fibonacci numbers indexed to the first two rotations of the spiral (a palindromic pattern {1393717997173931} that repeats every 16 Fibo index numbers) or, similarly,
decomposing and summing the prime root angles. The decomposition works as follows (in digit sum arithmetic this would be termed summing to the digital root): F17
(the 17th Fibonacci number) = 1597 = 1 + 5 + 9 + 7 = 22 = 2 + 2 = 4:
The thought plickens ...
Another dimension of symmetry involves the terminating digits of the prime roots and their angles:
those paired with like terminating digits being separated by 120°: 1(12°) and 11(132°) ... 13(156°)
and 23(276°) ... 7(84°) and 17(204°) ... 19(228°) and 29(348°). Another consideration with
regard to terminating digits, is that one can easily construct, by combining all numbers with the same
terminating digits, a four-fold arithmetic progression in increments of +10 and +20, starting with
1, 7, 13 and 19. Thus, combining 1(12°) and 11(132°) gives us: 1, 11, 31, 41, 61, 71, 91, [+10+20] ... n;
combining 7(84°) and 17(204°) gives us 7, 17, 37, 47, 67, 77, 97, [+10+20] ... n;
combining 13(156°) and 23(276°) gives us 13, 23, 43, 53, 73, 83, 103, [+10+20] ... n; and,
combining 19(228°) and 29(348°) gives us 19, 29, 49, 59, 79, 89, 109, [+10+20] ...n. Looking at the array in
this configuration, however, has borne no fruit.
As fascinating as the symmetries examined above may be, they are but a prelude to the beautiful patterns
we'll explore when we discuss digital root sequencing and the Trinity of Triangles and Magic Squares rooted in Vedic Arithmetic
that drive factorization algorithms within this domain. To go directly to this discussion click here.
The Croft Spiral Sieve
Around the perimeter of the spiral sieve, below, you'll note that the 8 radii are labeled in relation to their prime roots, i.e., 1(12°);
7(84°); 11(132°); 13(156°); 17(204°); 19(228°); 23(276°) and 29(348°). These relate to the fact that the circle
is segmented into 30 equal sectors or radii separated by 12° (30*12°=360°).
This sieve "exposes" the twin primes, aligning as they do along three distinct "distribution channels." One obvious
implication, is that those numbers in the array with mod 30,7 (radial angle 84°) and mod 30,23 (radial angle 276°) can be excluded
as twin prime candidates (and, by definition, all primes distributed along these two diagonals will be what are known as
"isolated primes"). Toward the bottom of this page
it is explained how the twin primes can be segregated from all other positive integers and be partitioned into three columnar sets covertly
aligned by the first three primes (encoded in angles).
Conjectures and Facts Relating to the Croft Spiral Sieve
The array is rooted in the first three primes: 2, 3 and 5 and their product, 30, the 3rd primorial. This array reveals that
the first three primes play a very special role in creating the symmetrical geometries that align the distribution of all
subsequent prime numbers, thus distinguishing them from all other primes. Primes 2, 3 and 5 are like 8-legged spiders assigned
to spin the beautiful spiraling web in which the remaining primes are arrayed along assigned
threads. (For a detailed listing of Number 30's attributes, plus reference links click on this link:
The Number 30). To tantalize you in the meantime, consider the
magic square, below, where every horizontal, vertical and corner-to-corner
diagonal sum totals 30, while the total for the horizontal and vertical sums combined = 360, making it, with 36 squares, a perfect
square of the first perfect number (6) squared in terms of number of squares, and when multiplied by 10 we have a perfect square
representing a circle. Explore its beautiful symmetries involving 10's derived from digit sum (aka Vedic)
arithmetic, the significance of which will become more evident when we reach our discussion of
Factorization Utilizing Digital Root Sequencing:
It is conjectured that all (and only) the numbers in this array (and their negatives) can be derived using the
interchangeable expression incorporating the first three primes, 2, 3 and 5, where x=2, y=3 and z=5.
Thus: xnyn ± zn, xnzn ± yn
and ynzn ± xn. For example: 2 * 3 + 5 = 11 ... 23 * 5 - 33 = 13 ...
32 * 5 + 2 = 47 ... 52 * 3 - 2 = 73. To see more examples (1 thru 101)
click here. This expression, therefore, potentially generates all numbers not divisible
by its three terms, 2, 3 and 5, including all primes >5. [Note: For any given number in the array, there are multiple–and
possibly an infinite number of–solutions. For example, the number 11 can be expressed as xy+z = 11,
x2y2-z2 = 11, z2y-x6 = 11, etc.]
All prime numbers (with the exception of 2, 3 and 5) are distributed along 8 diagonals in intervals of 30, starting with
"prime roots": 1, 7, 11, 13, 17, 19, 23 and 29 (thus: 1...31...61...91...n; 7...37...67...97...n; etc.).
The products of any combination of factors in the array = a number in the array, e.g.,
7*11 = 77; 7*11*13 = 1001; etc. Conversely, all factors for composite numbers in the array can be found in the array.
The sum of any sequential odd number of addends in the array = a number in the array, e.g.,
1+7+11 = 19; 1+7+11+13+17 = 49; etc.
Because the digital roots of all prime root angles are either 3 or 6, any prime root angle times another will produce
a product whose digital root = 9, e.g., PR7 (84°) x PR29 (348°) = 84 x 348 = 29232 = dr(9).
Any number in the array x 30 + 1 = a number in the array.
The sum of the angles for 2(24°), 3(36°) and 5(60°) = 120°, and the sum of the prime roots
(1+7+11+13+17+19+23+29) also = 120. This is because the prime roots are an arithmetic anagram for the angles
of the first three primes, thus: 11+13 = 24; 17+19 = 36; and 1+7+23+29 = 60. The sum of the second
rotation = 360 ... 3(2[24°] + 3[36°] + 5[60°]) = 30[360°]
The array reveals beautifully symmetrical relationships:
Mod 30 of all numbers in this array (and thus all primes other than 2, 3 and 5) must be 1, 7, 11, 13, 17, 19, 23
or 29. [Note: A shortcut for calculating the modulo 30 of any number in the set is as follows: First, determine the digital root of the number in question,
e.g., 49 = 4+9 = 13 (digit sum) = 1+3 = 4 (digital root). The numbers in this set reduce to 6 digital roots, grouped into two sets of 3, as follows:
{1,4,7} and {2,5,8}. The digital roots for a given modulo 30 repeat in intervals of 90, e.g., 31, 61, 91, 121, 151, 181 have digital roots of
4, 7, 1, 4, 7, 1 or {4,7,1 repeat}. When a number in the set's digital root is 1, 4 or 7, the number's modulo 30 will be 1, 7, 13 or 19. Match the
number's terminating digit with one of these four and you have your modulo 30. When a number in the set's digital root is 2, 5 or 8, its modulo 30
will be 11, 17, 23 or 29. Again, match up the number's terminating digit with one of these four and you have your modulo 30. For example, the number
3907 has a digital root of 1 and a terminating digit of 7; thus its modulo 30 is 7. In turn, the modulo 30 of 3707, which has a digital root of 8, will be 17.]
The sum of the digital root sums of the prime roots (1, 7, 11, 13, 17, 19, 23, 29) = 1+7+2+4+8+1+5+2 = 30.
This sieve reveals why all primes >5 are adjacent to a multiple of six, as the prime root radii are adjacent to 6(72°);
12(144°); 18(216°); 24(288°); and 30(360°). [And you'll note that the digital root sums of all adjacent angles equal 9.]
Factorization/Prime Number Sieving Methods
For a detailed discussion of efficient factorization and prime number sieving algorithms, as well as an in-depth analysis of
the 8-chord progression and deterministic modulo 90 digital root dyad sequences underlying all factorizations employing this sieve
click here.
Here's a sneak preview of the "magic square" you'll encounter if you click on the above link; a beautiful matrix that is the
beating heart of this body mathematical in that it functions as a digital root multiplication calculator that accounts for factors of all
composite numbers in our domain. And yes, the name "Imaginary Square" does have significance.
"Imaginary Square"
[Note: If you use Python (the computer programming language designed for the development of scientific, engineering and
mathematics applications) and want to cut to the chase, check out the MIT licensed Python module dubbed "pyprimes" designed to run and compare
prime number sieving algorithms, including the Croft Spiral Sieve, here.
The programmer, by-the-way, rates the Croft Spiral Sieve "effective and fast."]
Twin Primes
The twin primes are arrayed along three distinct "distribution
channels," i.e., each set of potential twins is separated by 24° along three sets of paired radii: [132° and 156°] ... [204° and 228°] ...
[348° and 12°]. Translated into modular arithmetic, mod 30 for the three twin prime candidate channels (A, B and C,
for convenience) distribute as follows, and in the sequences shown:
A: [n & n+2] mod 30 = [11, 13] B: [n & n+2] mod 30 = [17, 19] C: [n & n+2] mod 30 = [29, 1]
If you consolidate the 6 radii used to construct the twin prime distribution channels into a single number line, you can completely isolate
twin prime candidates, pairing them in tandem in a deterministic sequence, viz.: 11 {+2+4+2+10+2+10} {repeat ... ∞}.
For example, six of the first seven such pairings are twin primes: [11, 13] [17, 19] [29, 31] [41, 43] [47,
49 (72)] [59, 61] [71, 73]. Below is a matrix showing all the twin prime sets from [11, 13] to [569, 571]:
TWIN PRIME DISTRIBUTION MATRIX
11 {+2+4+2+10+2+10} {repeat ... ∞}
11
+2
13
+4
17
+2
19
+10
29
+2
31
+10
41
+2
43
+4
47
+2
49
+10
59
+2
61
+10
71
+2
73
+4
77
+2
79
+10
89
+2
91
+10
101
+2
103
+4
107
+2
109
+10
119
+2
121
+10
131
+2
133
+4
137
+2
139
+10
149
+2
151
+10
161
+2
163
+4
167
+2
169
+10
179
+2
181
+10
191
+2
193
+4
197
+2
199
+10
209
+2
211
+10
221
+2
223
+4
227
+2
229
+10
239
+2
241
+10
251
+2
253
+4
257
+2
259
+10
269
+2
271
+10
281
+2
283
+4
287
+2
289
+10
299
+2
301
+10
311
+2
313
+4
317
+2
319
+10
329
+2
331
+10
341
+2
343
+4
347
+2
349
+10
359
+2
361
+10
371
+2
373
+4
377
+2
379
+10
389
+2
391
+10
401
+2
403
+4
407
+2
409
+10
419
+2
421
+10
431
+2
433
+4
437
+2
439
+10
449
+2
451
+10
461
+2
463
+4
467
+2
469
+10
479
+2
481
+10
491
+2
493
+4
497
+2
499
+10
509
+2
511
+10
521
+2
523
+4
527
+2
529
+10
539
+2
541
+10
551
+2
553
+4
557
+2
559
+10
569
+2
571
+10
= Prime
= Interval
= Composite
prime
+2
prime
= Prime Pair
The first three twin prime sets in this matrix configuration are an anagram, in angular terms (in a 30-sectioned circle like the Croft Spiral Sieve),
to the first three primes, thus: [11, 13] [17, 19] and [29, 31] respectively translate to:
All twin primes > [5, 7] are thus partitioned into three groups (columns) by the omnipresent first three primes. [And you'll note that the
sum of the intervals for each row of twin prime candidates = 30.]
Another interesting aspect of the three twin prime distribution channels becomes apparent when the eight radials of the Croft Spiral Sieve (or modulo 30
wheel factorization), are superimposed upon a {10/3} regular star polygon (see illustration, below). Thirty (30)
is the total unit length of the ten straight line segments used to construct the {10/3} star polygon, where the straight lines connect every 3rd point of 10 equally spaced
points lying on a circle's circumference. Note that three points of the polygon, 144°(12), 216°(18) and 360°(30), precisely and symmetrically split the
twin prime distribution channels. And as you rotate this object, several other symmetries become apparent, e.g.: between 7 → 17; 13 → 23; and 7 → 23.
All composite numbers in the twin prime distribution channels can be accounted for by the eight-fold algorithmic progressions shown below. [Note:
The algorithm shown here is a sub-set of the chordal progressions discussed at some length earlier that factored all composite numbers in the
array populated by the set of all natural numbers not divisible by 2, 3 and 5, as opposed to only factoring those impacting twin prime candidates, shown below.]:
It is well known that the ratio of any Fibo(n+2)/Fibo(n) will converge to a limit φ + 1 = φ2 (an irrational number) as n approaches
infinity (φ being the symbol for phi or the "golden ratio"
which = 1.6180339887498948482045868343656 ...). And so it follows that the ratios of the Fibonacci numbers indexed to the twin primes (n and n+2) and/or
twin prime candidates in sequence converge accordingly (in other words, the square roots of the ratios of Fibonacci numbers indexed to the twin primes
and/or twin prime candidates in sequence converge to φ).
In sum, given that all composite numbers extant in the infinite set of natural numbers not divisible by 2, 3 and 5 (which by definition
contains all prime numbers ≥ 7 and their multiples and is defined as 1 {+6 +4 +2 +4 +2 +4 +6 +2} {repeat ... ∞}), can
be produced algorithmically, employing eight geometrically-expanding factorization sequences, ultimately leaving the equal distribution
of all prime numbers ≥ 7 ... n along eight diagonals; and given that six of these diagonals form the three twin prime distribution
channels (described above), which, when combined into a single number line, reveal a divergent series: 11 {+2 +4 +2 +10 +2 +10}
{repeat ... ∞}; and given that the infinity of prime numbers is proven, it's axiomatic that there is an infinity of prime pairs.
Put differently, prime pairs, starting with [11, 13] are sequenced within a divergent (aka harmonic) series while paradoxically converging
to φ2 when indexed to Fibonacci numbers like meshing gears.
One other twin prime related deterministic algorithm worth noting stair-steps its way up the twin prime
distribution channels with a periodicity of 6. We've dubbed these "perfect twins," given that the square root of their
sums is the first perfect number, 6, or one of its multiples. To calculate this interesting twin pair candidate sequence,
start with x = 6 (then add 6 and repeat for each successive step, as shown below):
_________________
√
(x2/2 - 1 + x2/2 +1)
x=6: 62 = 36; 36/2 = 18; 18-1 = 17; 18+1 = 19; thus, twin pair candidates = 17 and 19; take the square root of 17+19
which = 6
add 6 (6+6 = 12)
x=12: 122 = 144; 144/2 = 72; 72-1 = 71; 72+1 = 73; thus, twin pair candidates = 71 and 73; take the square root of 71+73
which = 12
add 6 (12+6 = 18) {repeat ... n}
The table below shows the first 20 steps in the sequence. All "perfect twins" are hi-lited in gold.
[Note: The table is arrayed in five columns to show the +30 and repeating modulo 30 vertical incrementation. Also note that the digit
root sums for all prime pair candidates and/or prime pairs in this sequence = 9, e.g., 17+19 = 36; 3+6 = 9; ... 647+649 = 1296;
1+2+9+6 = 18; 8+1 = 9 ... 1151+1153 = 2304; 2+3+0+4 = 9; 6497+6499 = 12996; 1+2+9+9+6= 27; 2+7 = 9; etc.]
√ (17+19) = 6
√ (71+73) = 12
√ (161+163) = 18
√ (287+289) = 24
√ (449+451) = 30
↑ Mod30, 17 & 19 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 17 & 19 ↑
↑ Mod30, 29 & 1 ↑
√ (647+649) = 36
√ (881+883) = 42
√ (1151+1153) = 48
√ (1457+1459) = 54
√ (1799+1801) = 60
↑ Mod30, 17 & 19 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 17 & 19 ↑
↑ Mod30, 29 & 1 ↑
√ (2177+2179) = 66
√ (2591+2593) = 72
√ (3041+3043) = 78
√ (3527+3529) = 84
√ (4049+4051) = 90
↑ Mod30, 17 & 19 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 17 & 19 ↑
↑ Mod30, 29 & 1 ↑
√ (4607+4609) = 96
√ (5201+5203) = 102
√ (5831+5833) = 108
√ (6497+6499) = 114
√ (7199+7201) = 120
↑ Mod30, 17 & 19 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 11 & 13 ↑
↑ Mod30, 17 & 19 ↑
↑ Mod30, 29 & 1 ↑
Before leaving the twin primes, let's take a look at them from a digital root perspective. The illustration
below shows the infinitely repeating digital root sequences for each of the 8 radii of the spiral sieve, as well the sums of
the digital roots (in yellow) between each radius, including the twin prime distribution channels. It's notable that the twin
pair digital sum sequences, {3,9,6}, {9,6,3} and {6,3,9} when positioned adjacent to each other in ascending order, create a
"magic square" (in that all vertical, horizontal and corner-to-corner diagonal sums equal 18), as pictured below:
Counting Primes
Although not particularly straightforward, there is a method whereby one can calculate the number of primes
in a given range with considerable precision using this Sieve. To illustrate, below is a step-by-step
procedure to determine the number of primes in the range 1 thru 10,000 [Note: Please consider this section to be more conceptual
outline than final product in as much as the author will not be satisfied until the process is drastically improved.]:
First, calculate the number of integers within the range not divisible by 2, 3 and 5. Since 8 out of every 30 integers
fall into this category, we have 8/30 = .2666666666... or 26.66%. Multiplying 10,000 times .266666... gives us
2666. (This is our starting point. The steps that follow will identify the number of non-primes to subtract until we've boiled
2666 down to reflect only primes.)
Next, given that the square root of 10,000 is 100, we know that we need to perform chord factorizations from 7chord to
97chord to generate products (i.e., non-primes) less than or equal to 10,000 (reference "The 8-Chord Progression,"
above). Doing so will index all composite numbers in this array (aside
from those divisible by 2, 3 and 5, already removed in step 1).
We then count the number of calculations required to complete step 2, above. This gives us 1847. (To open
an Excel spreadsheet showing in detail how this number is derived, click here. This
may take several seconds!)
Because the factorization process generates a number of duplicate products, we must count and subtract these
from the total in step 3. [Note: These duplications were described earlier, and the example given is repeated
here: 7*77 = 539 is equivalent to 11*49 = 539; both expressions being equivalent to 7*7*11 = 539.]. In our example, we find
that there are 411 such duplications; subtracting these from the number of factorizations (1847-411) gives us 1436.
To open an Excel spreadsheet listing the duplicates and their count, click here.]
Next, we subtract the result in step 4 (1436) from the result in step 1 (2666), i.e., 2666-1436 = 1230.
The final step is to subtract 1 from the total in step 5 to account for the fact that the number 1 is not a prime; this
leaves us with a balance of 1229, which is exactly the number of primes between 1 and 10,000.
No doubt this procedure can be simplified. For example, one could probably devise formulas to
determine the number of factorizations and duplicates, eliminating the need for a spreadsheet count.
(The author's next project :-).
Distribution of Squares
The squares of all numbers in the array distribute along two–and only two–prime root radii: 1(12°) and 19(228°).
It thus follows that the squares of all primes >5 can only be found distributed along prime root radii 1(12°) [mod30 = 1] and
19(228°) [mod30 = 19]. This is consistent with the fact that the quadratic residues
for modulo 30 (making them congruent with perfect squares) are 1 and 19. Also, this can be validated with digit sum arithmetic, given that
a) All perfect squares end in 1, 4, 5, 6, 9 or 0, only two of which (1 and 9) apply to our domain, i.e., when mod30 = 1, 11, 19 or 29; and
b) All perfect squares possess digital roots of 1, 4, 7 or 9, which in the case of our domain narrows the possibilities to mod30 = 1 or 19, as
all mod30 11 and 29 elements possess digital roots of 2, 5 or 8. [And it follows that all squares in this series distribute evenly to two of the
three twin prime distribution channels, described above, negating a significant percentage of potential prime pairs.] The matrix below illustrates
the distribution of squares from 1*1 thru 59*59 (squares hi-lited in blue):
12°
84°
132°
156°
204°
228°
276°
348°
1
7
11
13
17
19
23
29
A
B
C
D
E
F
G
H
AA (1*1)
AB (1*7)
AC (1*11)
AD (1*13)
AE (1*17)
AF (1*19)
AG (1*23)
AH (1*29)
BD (7*13)
BA (7*31)
BG (7*23)
BF (7*19)
BC (7*11)
BB (7*7)
BH (7*29)
BE (7*17)
CC (11*11)
CE (11*17)
CA (11*31)
CG (11*13)
CB (11*37)
CH (11*29)
CD (11*13)
CF (11*19)
DB (13*37)
DF (13*19)
DE (13*17)
DA (13*31)
DH (13*29)
DD (13*13)
DC (13*41)
DG (13*23)
EG (17*23)
EC (17*41)
ED (17*43)
EH (17*29)
EA (17*31)
EE (17*17)
EF (17*19)
EB (17*37)
FF (19*19)
FD (19*43)
FH (19*29)
FB (19*37)
FG (19*23)
FA (19*31)
FE (19*47)
FC (19*41)
GE (23*47)
GH (23*29)
GB (23*37)
GC (23*41)
GF (23*49)
GG (23*23)
GA (23*31)
GD (23*43)
HH (29*29)
HG (29*53)
HF (29*49)
HE (29*47)
HD (29*43)
HC (29*41)
HB (29*37)
HA (29*31)
12°
84°
132°
156°
204°
228°
276°
348°
1
7
11
13
17
19
23
29
A
B
C
D
E
F
G
H
AA (31*31)
AB (31*37)
AC (31*41)
AD (31*43)
AE (31*47)
AF (31*49)
AG (31*53)
AH (31*59)
BD (37*43)
BA (37*61)
BG (37*53)
BF (37*49)
BC (37*41)
BB (37*37)
BH (37*59)
BE (37*47)
CC (41*41)
CE (41*47)
CA (41*61)
CG (41*53)
CB (41*67)
CH (41*59)
CD (41*43)
CF (41*49)
DB (43*67)
DF (43*49)
DE (43*47)
DA (43*61)
DH (43*59)
DD (43*43)
DC (43*71)
DG (43*53)
EG (47*53)
EC (47*71)
ED (47*73)
EH (47*59)
EA (47*61)
EE (47*47)
EF (47*49)
EB (47*67)
FF (49*49)
FD (49*73)
FH (49*59)
FB (49*67)
FG (49*53)
FA (49*61)
FE (49*77)
FC (49*71)
GE (53*77)
GH (53*59)
GB (53*67)
GC (53*71)
GF (53*79)
GG (53*53)
GA (53*61)
GD (53*73)
HH (59*59)
HG (59*83)
HF (59*79)
HE (59*77)
HD (59*73)
HC (59*71)
HB (59*67)
HA (59*61)
From the above you can see that all squares distribute as follows: A*A (translated as any squared mod 30,1 number in the array will produce a mod 30,1 product),
C*C, F*F and H*H distribute to A (mod 30,1..12°), while B*B, D*D, E*E and G*G distribute to F (mod 30,19..228°).
From Alpha to Omega
The Ulam Spiral arrays primes in fragmented spiral and diagonal formations.
Quoting from Wikipedia: "Since in the Ulam spiral adjacent diagonals
are alternatively odd and even numbers, it is no surprise that all prime numbers lie in alternate diagonals ... What is
startling is the tendency of prime numbers to lie on some diagonals more than others." From this one
might deduce that the Ulam Spiral is very likely a scrambled version of the Croft Spiral Sieve as the latter demonstrates
how all prime numbers (except 2, 3 and 5) are fundamentally arrayed along eight (and only eight) diagonals.
It is conjectured that the Croft Spiral Sieve is congruent with the most complex and visually arresting
Lie group,
named E8, which–like
the Croft Spiral Sieve–is 8-dimensional (E8 is pictured below superimposed with a star polygon and the 8 radii of
the modulo 30 factorization wheel). This group was recently in
the news as possibly being a key to unifying theories in gravity and particle physics to create the proverbial "theory of everything."
The number 30–integral to the Croft Spiral Sieve–is the Coxeter Group
number h, dual Coxeter number and the highest degree of fundamental invariance of E8. You'll note, looking
at the graphical representation of E8 below, that the perimeters of every one of its multiple concentric circles possesses
30 points. And, not surprisingly, E8 has 2-, 3- and 5-torsion and its exponents are the co-primes up to 30,
i.e., 1, 7, 11, 13, 17, 19, 23, and 29–numbers you're very familiar with if you've read to this point ... which brings us
full circle Ο:
The Number 30 in Lie Group E8: This graphic superimposes images of the star
polygon, modulo 30 wheel factorization radii and "E8 graph of the Gosset 421 polytope as a 2-dimensional skew orthogonal projection
inside Petrie polygon ... an emulation of the hand drawn original by Peter McMullen" licensed by Creative Commons;
license terms here. (For 3-D version of this graphic step back three feet from
the screen.)
