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Your method is generating the set of natural numbers not divisible by 2, 3 or 5 (aka numbers congruent to {1, 7, 11, 13, 17, 19, 23, 29} modulo 30), which I explore in great depth employing the Prime Spiral Sieve (a spiraling 8-dimensional modulo 30 wheel factorization algorithm that accounts for all prime numbers >5 and their multiplicative multiples, starting with 7^2 -- which is why 49 is the first composite number in the string you've created, followed by 7 x 11=77 ... the 2nd composite number in your sequence, etc.). The algorithm in question (usually called the "Croft Spiral") has been written in several programming languages and is widely used for efficient deterministic (non-probabilistic) prime factorization. Check it out at http://www.primesdemystified.com.
Yeah, somehow my comment got posted to the wrong string. I thought I was somewhere else in this labyrinth :-).
Sorry 'bout that. Thanks for catching, Land of Tomorrow! btw, I've subsequently published my 'outline for a proof' at http://www.primesdemystified.com/twinprimesdigitalrootproof ... I copied the link, so the spelling s/b correct :-) Thanks again.
Employing a modulo 30 factorization wheel as an analytical tool, coupled with deductive logic, I believe one can prove that all twin primes other than (3,5) or (5,7) must have digital root sequencing of either {2,4}, {8,1} or {5,7}, i.e., there can be no twin primes that sequence with digital roots {7,1}. Though not stated explicitly, I believe one can find all the elements needed to construct a proof at primesdemytified.com/twinprimes. Regardless, here is an outline of the steps I would propose:
1. Narrow the field to natural numbers ≡ to {1,7,11,13,17,19,23,29} modulo 30 (oeis.org/A007775). This infinite sequence is commonly known as “numbers not divisible by 2, 3 or 5,” and by definition contains all prime numbers > 5 and their multiplicative multiples. Thus we exclude prime numbers 2, 3 and 5 from our analysis, and the two pairs of twin primes that 3 and 5 are members of: (3,5) and (5,7).
2. Populate a modulo 30 factorization wheel with the above-defined sequence where 1 = 12°. Upon doing so, numbers ≡ to {1,7,11,13,17,19,23,29} modulo 30 are evenly distributed along 8—and only 8—radii: {12°, 84°, 132°, 156°, 204°, 228°, 276°, 348°} and have a repetition cycle of 1{+6+4+2+4+2+4+6+2} {repeat…}. Each rotation around the wheel increments a difference sequence of +30 (for example, numbers ≡ {1} modulo 30 increment as follows: 1, 31, 61, 91, 121, 151, 181 … ). And therefore, in digital root terms, every rotation around the wheel increments each of the 8 radii by +3.
3. Upon completing Step 2, we see that the twin prime candidate field can be further reduced given that numbers radiating at 84° (or numbers ≡ {7} modulo 30) and 276° (or numbers ≡ {23} modulo 30) cannot possibly be twin primes given the closest prime numbers in proximity to numbers ≡ {7} modulo 30 is +4 and for numbers ≡ {23} modulo 30 is -4. We have now winnowed the field of twin prime candidates to numbers ≡ {1,11,13,17,19,29} modulo 30 (oeis.org/A230462) which has a repetition cycle of 11{+2+4+2+10+2+10+2} {repeat…}. The sequence that remains, when consolidated into a single number line, accounts for exactly 20% of natural numbers and consists exclusively of twin prime candidates (n, n+2), starting with (11,13).
4. We now note, given that each rotation around our sieve increments 8 elements +30 {repeat …}, the digital root sequencing of each radius increments by digital root +3. Thus, the digital root sequencing of the 8 radii have a deterministic dependence upon the initial digital root state of the first 8 elements of the sequence where 1, 7, 11, 13, 17, 19, 23, 29 translates to digital roots 1, 7, 2, 4, 8, 1, 5, 2. The 8 radii thus sequence as:
numbers ≡ {1} modulo 30 sequence as {1,4,7} (or 1+3=4; 4+3=7; 7+3=1; repeat ... you get the drill ...)
numbers ≡ {7} modulo 30 sequence as {7,1,4}
numbers ≡ {11} modulo 30 sequence as {2,5,8}
numbers ≡ {13} modulo 30 sequence as {4,7,1}
numbers ≡ {17} modulo 30 sequence as {8,2,5}
numbers ≡ {19} modulo 30 sequence as {1,4,7}
numbers ≡ {23} modulo 30 sequence as {5,8,2}
numbers ≡ {29} modulo 30 sequence as {2,5,8}
5. As stated in Step 3, for the purposes of examining twin prime digital root sequencing, we can ignore numbers ≡ {7, 23} modulo 30, leaving us with numbers ≡ {1,11,13,17,19,29} modulo 30, and their corresponding deterministic digital root sequencing:
numbers ≡ {1} modulo 30 sequence as {1,4,7}
numbers ≡ {11} modulo 30 sequence as {2,5,8}
numbers ≡ {13} modulo 30 sequence as {4,7,1}
numbers ≡ {17} modulo 30 sequence as {8,2,5}
numbers ≡ {19} modulo 30 sequence as {1,4,7}
numbers ≡ {29} modulo 30 sequence as {2,5,8}
When we pair these as n, n+2 dyads (twin prime candidate couplings) we see that:
Numbers ≡ {11,13} mod 30 sequence as:
(11,13) = digital roots {2,4}
(41,43) = digital roots {5,7}
(71,73) = digital roots {8,1}
{digital roots repeat …}
Numbers ≡ {17,19} mod 30 sequence as:
(17,19) = digital roots {8,1}
(47,49) = digital roots {2,4}
(77,79) = digital roots {5,7}
{digital roots repeat …}
Numbers ≡ {29, 1} mod 30 sequence as:
(29,31) = digital roots {2,4}
(59,61) = digital roots {5,7}
(89,91) = digital roots {8,1}
{digital roots repeat …}
And thus we conclude that all twin prime candidates of the form n, n+2 greater than (5,7) that are potentially p, p+2 distribute to one of three digital root dyadic sequences: {2,4} (oeis.org/A232880), {8,1} (oeis.org/A232882) or {5,7} (oeis.org/A232881) (and you’ll note that these decompose to {6,9,3}, the vertices of one of three rotating equilateral triangles, along with {1,4,7} and {2,5,8}, that interact to form tensor matrices representing the coordinates of {9/3} star polygons … but that’s another story …
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