On The Infinity of Twin Primes and other K-tuples

On The Infinity of Twin Primes and other K-tuples Jabari Zakiya: jzakiya@gmail.com April 4, 2024 Abstract The paper uses the structure and math of Prime Generator Theory to show there are an infinity of twin primes, proving the Twin Prime Conjecture, as well as establishing the infinity of other k-tuples of primes. 1 Introduction In number theory Polignac’s Conjecture (1849) [6] states there are infinitely many consecutive primes (prime pairs) that differ by any even number n. The Twin Prime Conjecture derives from it for prime pairs that differ by 2, the so called twin primes, e.g. (11, 13) and (101, 103). K-tuples are groupings of primes adhering to specific patterns, usually designated as (k, d) groupings, where k is the number of primes in the group and d the total spacing between its first and last prime [4]. Thus, Polignac’s pairs are type (2, n), where n is any even number. Three named (2, n) tuples are Twin Primes (2, 2), Cousin Primes (2, 4), and Sexy Primes (2, 6). The paper shows there are many more Sexy Primes (in fact, always more abundant) than Twins or Cousins, though an infinity of each, and an infinity of any other (2, n) tuple. I begin by presenting the foundation of Prime Generator Theory (PGT), through its various components. I start with Prime Generators (PG), which as their name implies, generate all the primes. Each larger PG is more efficient at identifying primes by reducing the number space they can possibly exist within. They thus structurally squeeze the primes into a smaller set of integers that contain fewer composites, in a very systematic manner. Each PG has a characteristic Prime Generator Sequence (PGS), a repeating pattern of gaps between the residue elements of its PG. These gap patterns illustrate, and adhere to, a deterministic set of properties. I use them to systematically show once a PGS gap size between residues exists it will be repeated with higher frequency for all larger PGS. I then show every residue gap will, with certainty, become a gap strictly between prime pairs. This will be used to establish the infinity of twin pairs, and other k-tuples. I provide data and graphs to empirically show this. The epistemological model for developing PGT is highly visual, and most easily explained and understood through pictures to establish its properties. Some may not find this “rigorous” and insufficient to meet its claims. However, it will be seen its foundation provides a consistent mathematical framework to qualitatively explain, and quantitatively produce, empirically verifiable results derived using other methods and techniques. At the time of writing, the largest known twin prime is 2996863034895 · 21290000 ± 1 [5] (2016), which resides on restracks P5[29:31] and P7[29:31] for those PG. There are an infinity of larger 1 twin primes, which will reside on some twin pair restracks for every PG. The same will be true for other k-tuples. I have previously used Prime Generators to construct and implement efficient and very fast prime sieves, to find all the primes up to a finite N, or within a finite range, including the fastest and most efficient prime sieve methods to find all primes and twin|cousin primes. See [1], [2], [3] 2 Prime Generators A prime generator Pn is composed of a modulus modpn and a∏set of residues r i with residue count rescntpn (determined by Euler’s Totient Function, ϕ(n)= n (1 − 1/pi ), which have the form: Pn = modpn · k + {ri } modpn = pn # = rescntpn = (pn − 1)# = ∏ ∏ pi = 2 · 3 · 5 · ... · pn (pi − 1) = (2 − 1) · (3 − 1) · (5 − 1) · ... · (pn − 1) (1) (2) (3) where pn is the last PG prime. A PG’s residues are the set of integers r i ε {1...modpn-1} coprime (no common factors) to its modpn, i.e. their greatest common divisor is 1: gcd(ri , modpn) = 1. They exist as modular complement pairs, such that modpn = r i + r j and therefore (r i + r j ) mod modpn ≡ 0. Thus, we only need to generate the residues r i < modpn/2, and the other half are r j = modpn - r i . For P5 then, modp5 = 2 · 3 · 5 = 30, with rescntp5 = 1 · 2 · 4 = 8. P5’s 8 canonical residues are {1, 7, 11, 13, 17, 19, 23, 29}, which are used functionally as {7, 11, 13, 17, 19, 23, 29, 31}, to always have the first residue in the sequence be prime pn+1 , and permute ri = 1 to be the last residue in the sequence, set to (modpn + 1) ≡ 1 mod modpn. Thus we have: P 5 = 30 · k + {7, 11, 13, 17, 19, 23, 29, 31} (4) We can now construct P5’s prime candidates (pcs) table, here up to N = 541, the 100th prime, where each k ≥ 0 index residue group (resgroup) contains pc values along each residue track (restrack|rt). Fig 1. 2 A table of prime candidates can be created for every PG. All the primes > pn occur mostly in equal numbers (i.e. statistically uniformly) along each restracks. The marked cells in Fig 1. are prime multiples (composites) of the residue primes, that have been sieved out to identify the primes within the range. See [1], [3]. P5 is the largest Pn for which all its residues are prime. All larger will have residues consisting of primes and their consecutive coprime multiples < modpn. 3 Prime Generator Sequences Each prime generator has a characteristic Prime Generator Sequence (PGS). This is the sequence of the differences (gaps) between consecutive residues defined over the range r 0 to r 0 + modpn where r 0 is the first residue of Pn, which is the next prime > pn , i.e. pn+1 . Let’s construct the first prime generator P2, and its PGS. For P2: modp2 = 2, with rescntp2 = (2 - 1) = 1, with residue {1}, but use its functional value {3}. Thus, P2 = 2·k+3, produces the pc sequence: 3 5 7 9 11 13 15 17... ∞, i.e the odd numbers. So for P2, its PGS is a single element of gap size (r 0 - 1) = (3 - 1) = 2: PGS P2: [r 0 = 3] 2 | Now let’s construct P3: modp3 = 2 · 3 = 6; rescntp3 = (2 - 1) · (3 - 1) = 2, with residues {1, 5}. P3, thus, has the functional form: P3 = 6·k+ {5, 7}. Its pcs table is shown below up to k = 16. k rt0 rt1 0 5 7 1 11 13 2 17 19 3 23 25 4 29 31 5 35 37 6 41 43 7 47 49 8 53 55 9 59 61 10 65 67 11 71 73 12 77 79 13 83 85 14 89 91 15 95 97 16 101 103 Fig 2. For P3, each resgroup (column) contains prime candidates forming a possible twin pair, extending into infinity. Except for (3, 5), every twin prime can be written as 6n ± 1 for some n ≥ 1 values. The last two residues for all prime generators > P2 are modpn ± 1, thus they have at least one twin pair set of residues. For larger prime generators there are more twin pair residues, and others. To illustrate this, we examine the PGS for increasing prime generators Pn. For P3 we see its PGS contains the gaps 2 and 4, which occur one each, with the last (r0 - 1) = 4. PGS P3: 5 7 11 13 17 19 23 25 29 31 35 . . . ∞ 2 4| 2 4| 2 4| 2 4| 2 4| For P5 we see from Fig 1. its sequence of prime candidates, with its PGS spacing. PGS P5: 7 11 13 17 19 23 29 31 37 41 43 47 49 53 59 61 67 . . . ∞ 4 2 4 2 4 6 2 6| 4 2 4 2 4 6 2 6| Again we see the gaps 2 and 4 occurring with the same (odd) frequency, with the last three gaps now having the form (r 0 - 1) 2 (r 0 - 1), where r 0 = 7 is the first residue for P5. We are beginning to see some of the inherent properties of prime generators emerge. Each larger Pn (P7, P11, P13, P17, etc) will conform to these properties, producing an increasing number of gaps, with a defined number of specific gap sizes, systematically distributed within the sequence. 3 4 Characterizing PGS Each prime generator sequence is defined over the range r 0 to r 0 + modpn, therefore the number of gaps equals the number of residues, and the sum of the gap sizes equals the modulus. Let ai be the frequency coefficients (number of occurrences) for each gap of size 2i, i ≥ 1, thus: ∑ rescntpn = ai (5) modpn = ∑ gapi = ∑ ai · 2i (6) Therefore for PGS P3: [r0 = 5] 2 4 | → modp3 = 6 = (1) · 2 + (1) · 4 and PGS P5: [r0 = 7] 4 2 4 2 4 6 2 6 | → modp5 = 30 = (3) · 2 + (3) · 4 + (2) · 6 For P7, modp7 = modp5 · 7 = 210, and rescntp7 = rescntp5 · (7 - 1) = 48, with the residues: {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, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211} PGS P7: [r0 = 11] 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2 4 2 4 8 6 4 6 2 4 6 2 6 6 4 2 4 6 2 6 4 2 4 2 10 2 10 | With: modp7 = 210 = (15) · 2 + (15) · 4 + (14) · 6 + (2) · 8 + (2) · 10 Again we see for P7, there are an equal odd number of occurrences for gaps 2 and 4. This illustrates a property of every prime generator with modulus of pn #, coefficients a1 = a2 have form: ∏ a1 = a2 = (pn − 2)# = (podd − 2) = (3 − 2) · (5 − 2) · (7 − 2) · ... · (pn − 2) (7) We also see the consistent pattern that the last gap term is (r 0 - 1), and starting with P5, the last three gaps have the pattern (r 0 - 1) 2 (r 0 - 1). This occurs because the last two residues are always twin pairs of form modpn ± 1, and the second from last is the modular complement of r 0 , i.e. (modpn - r 0 ). We now also notice that the number of unique gap sizes for each generator Pn are of order pn−1 . This is observed to be the minumum number of gaps for increasing Pn (for nonzero coefficients). Thus the PGS for P3 has two (2) gaps, for P5 three (3) gaps, for P7 five (5) gaps sizes, and so on. 5 PGS Symmetry and Distribution Because the residues exist as modular complement pairs they produce a mirror image gap distribution around a midpoint pivot term. The PGS pattern up to the pivot will exist as its mirror image after. Starting with P5, we know the last 3 gaps for all Pn have the form (r 0 - 1) 2 (r 0 - 1), thus their sum is 2r 0 , and the remaining odd number (rescntpn - 3) gaps must equal (modpn - 2r 0 ). 4 This requires for P5, the (8 - 3) = 5 gaps at the front of its PGS must sum to (30 - 2·7) = 16. If all the gaps were 2 you would need 8, which is too many, if all were 4 you need just 4, which is too few. The gap structure is numerically constrained to generate the unique combination of gap sizes to satisfy both requirements (5) and (6) that represent each Pn. In addition, these (rescntpn - 3) odd gaps exist with a symmetric mirror image distribution around a mid pivot gap that is always of size 4 for pn # moduli. To show this, excluding the last 3 term of PGS P5 we have the gap sequence: 4 2 4 2 4 Here the terms 4 2 are the mirror image of 2 4 and are symmetric around midterm 4. For PGS P7 we get: 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2 4 2 4 8 6 4 6 2 4 6 2 6 6 4 2 4 6 2 6 4 2 4 2 and again see a similar mirror image symmetry of each half around the midterm 4. For P7, in order for the (48 - 3) = 45 gaps in its PGS front to sum to (210 - 2·11) = 188 we see new gaps of 8 are introduced (mirrored in both halves) close to the middle pivot point. As the PG moduli increase, new larger gaps will emerge and be included toward the pivot element. This amounts to pushing the preexisting gaps toward the front and back. This expansion process ensures all preexisting residue gaps will eventually exist for the primes < r 0 2 for some Pn. Each PGS shows a1 = a2 are odd because gap size 4 is the pivot term and a gap 2 is part of the last three sequence terms. (I provide the numerical basis for this in the Appendix.) Every other gap term is part of each mirror image and therefore occur in even numbers. Thus as similar to the residues, we only need to (computationally) determine the first (rescntpn - 4)/2 gap terms. 6 The Infinity of Primes Starting with just the first two primes 2 and 3, we can show the infinite progression of primes. Using the first two primes we create: P3 = 6·k + {5, 7}, k ≥ 0. From Fig 2. the pcs < r 0 2 = 52 = 25 are prime, which are the values {5, 7, 11, 13, 17, 19, 23}. We now use the new found primes 5. . . 23 to construct P23, with modp23 = 223092870, whose r 0 = 29. All the residues between 29 and 292 = 841 will be primes. The primes counting function π(x) tells us there are exactly 137 primes from 29...841, the last being 839. We now have a repeatable deterministic process to identify all the primes, into infinity . Thus, any prime p can be treated as r 0 to a Pn modulus composed of all the primes < p, whose residues from p to p2 are new primes. We can repeat this progression of primes process forever, to always generate new primes. Thus from this exact process, we can generate a list of consecutive primes for any Pn, from which we can then exactly determine their prime gaps distribution. In fact, an estimate of the number of new primes generated in any range p to p2 will be of order: π est (p, p2 ) = p2 p p · (p − 2) − = 2 log(p ) log(p) 2 · log(p) For p = 29, this produces an estimate of 116 primes from 29 to 841, compared to the actual of 137. (See Appendix for fuller elaboration.) 5 (8) 7 Prime Generator Properties Given what we’ve observed, and now know about prime generators and their sequences, we can codify their inherent and immutable properties, and use them in a logically consistent manner to empirically establish and project the nature, numbers, and distribution of all prime gap k-tuples. Though mathematically simple expressions, prime generators reveal an astounding breadth of knowledge about the nature of prime numbers, embedded in their inherent immutable properties. When I refer to their properties as being ‘inherent’ these are natural aspects and characteristics of their structure that are discernible easily through visual observation. Once observed they could be mathematically described and characterized to formulate a consistent framework for application. As an example, it is an inherent property of base ten numbers that the least significant digit (lsd ) of an even integer must (only) be the digits, 0, 2, 4, 6, 8, and conversely 1, 3, 5, 7, 9 for odd. However when we change the base system, say to a binary (base two) system, even|odd has a different expression, i.e. the least significant bit (lsb) of an even number is a ‘0’ and a ‘1’ for odd. We performed no calculation to determine this, these are observable characteristics that are inherently associated with the concepts of even and odd for each base system. Using these inherent properties of even|odd for base ten numbers, we can apply them through observation to ‘prime’ numbers. It is an inherent property of prime numbers that, other than for the prime 2, all others are odd, which means their lsd aren’t 0, 2, 4, 6, or 8. So by mere observation you know 341786 isn’t prime. You didn’t need to perform a calculation to confirm this, if you understood this natural inherent property of prime numbers it’s observably obvious. Also, other than for the prime 5, all other primes lsd can only be 1, 3, 7, or 9. This means at minimum 60% of all integers (those with lsd of 0, 2, 4, 5, 6, and 8) can’t be primes. This is an inherent property of numbers. If you know a little bit more number theory, you also know that while 11 and 101 could be primes (they are) 111, 1011, and 1101 observably could not. Why? Because for base ten numbers, if the sum of their digits is a multiple of 3 then it’s divisible by 3, and thus not prime. Thus it is an inherent property of Twin Primes their lsd can only be {1, 3}, {7, 9}, or {9, 1} e.g. for (11, 13), (17, 19), and (29, 31). It’s also inherent for all prime numbers > 2, the gaps between them are even because each is odd. You don’t have to ‘prove’ this (though the proof is simple), it is an inherent property of odd numbers. Thus, when I refer to the inherent properties of prime generators, these are observable characteristics and patterns that emerge naturally from their structure which I have mathematically codified. They are also immutable because they are the same for all generators constructed as shown, and can’t change. Constructing the Pn modulus as the primorial of primes pn totally determines its structure, as the residues count is determined by the Euler Totient Function, their values by the gcd test, and the residue values determine their gap sizes, whose distribution is determined by the symmetric properties of their modular forms. There is nothing random in this process. So while there is a clear deterministic numerical foundation for PGT, visualization of its elements reveal and explains it best. You have to draw pictures, e.g. Fig 1. and generator sequences, and produce enough examples to visually reveal their patterns. You cannot imagine these properties into existence just from numerical analysis, you have to observe them first. Now that I have described and given examples of prime generators and their sequences, I will list their observable inherent properties, which I have codified into a mathematically consistent framework for application. 6 Major Properties of Prime Generators • the modulus of every prime generator with last prime pn has primorial form: modpn = pn # • the number of residues are even with form: rescntpn = (pn - 1)# • the residues occur as modular complement pairs to its modulus: modpn = ri + rj • the last two residues of a generator are constructed as: (modpn - 1) (modpn + 1) • the residues include all the coprime primes up to modpn • the first residue r0 is the next prime > pn • the residues from r0 to r 0 2 are primes • each prime generator has a characteristic sequence of even sized residue gaps • the last 3 sequence gaps have form: (r0 - 1) 2 (r0 - 1) • the gaps are distributed in a symmetric mirror image around a pivot gap of size 4 • the residue gaps sum from r0 to r0 + modpn equals the modulus: modpn = Σai · 2i • the coefficients ai are the frequency of each gap of size 2i • the sum of the coefficients ai equal the number of residues: rescntpn = Σai • coefficients a1 = a2 are odd and equal with form: a1 = a2 = (pn - 2)# • the coefficients ai are even for i > 2 • the minumum number of nonzero coefficients ai in the sequence for Pn is of order pn−1 These inherent and immutable properties form a bounded set of constraints which characterize the formation and distribution of primes, and thus also the distribution of all their prime k-tuples. These discrete mathematical properties and operations form a striking correlation to calculus, where for distance x(t) its first derivative is velocity = dx(t)/dt and its second derivative is acceleration = dv(t)/dt. For prime generators, distance is the number span covered by modpn, and its derivative are the number of residues|gaps. Taking the derivative of the number of gaps gives us the actual gap size coefficients. Calculus Prime Generators ∏ x(t) = S v(t)dt modpn = Σai ·2i = pi ∏ v(t) = S a(t)dt rescntpn = Σai = (pi - 1) ∏ a(t) = S A(t)dt a1 = a2 = (pi - 2) While calculus integration is analogous to discrete summation, it is not intuitive that discrete summation correlates to primorial operators for prime generators. Or is it? Actually we see a similar relationship with the Riemann Zeta series and its equivalent Euler primes product form. ∏ s ∑ 1 ∏( ) modps p −s −1 ∏ ⇒ = 1 − p = (ps − 1) ns rescntps 7 (9) 8 Proof of The Infinity of Twin Primes and other k-tuples Theory of Proof For every Pn with largest modulus primorial prime pn , its residues contain the consecutive primes pi from r 0 ≤ pi ≤ pn # + 1, and their coprime composites, whose total is (pn − 1)#. In general, we don’t know which residues are primes over the whole range. However, if we limit the range of interest to r 0 to r 0 2 we know those residues are consecutive primes (as r 0 = pn+1 is the first prime > pn , the residues from pn+1 to p2n+1 are the consecutive primes > pn and < p2n+1 coprime to pn #). Thus the gaps between these prime residues constitute the distribution of their prime pair k-tuples. Since we know the residue gap distribution over the whole range, we can estimate with high accuracy their distribution in this range. We find as the residue gaps increase in size and frequency as pn increases, the prime gaps from pn+1 to p2n+1 similarly increase, for any gap size n as pn → ∞. Thus, for the infinity of residue gaps sizes n there are an infinity of (2, n) prime tuples. Thus the simplest and elementary proof of the infinity of k-tuples establishes their endless progression in the range r 0 to r 0 2 , for as Pn increases: 1) the residue gaps coefficients ai (for gap sizes n = 2i) increase for size and frequency, without end, and 2) as there are an infinity of r 0 = p primes, and ranges p to p2 , they will contain an increasing number of prime pairs for any gap size n, without end, as pn → ∞. We start by noting again for all Pn: ∑ modpn = pn # = ai · 2i (10) rescntpn = (pn − 1)# = ∑ ai (11) Proposition 1. As Pn increases, residue gap coefficients ai increase infinitely in size and frequency. Proof. From (11) as modpn increases by pn the number of residues increase by (pn − 1), which equal the number of residue gaps. From (10) we also know the sum of occurrences for each gap size equals the modulus value. The smaller ai gaps occur first, and in highest frequency, as a function of increasing pn , while larger gap sizes ak are functions of the smaller ones, and also systematically increase in frequency with pn . Thus as Pn increases by pn , the number of unique residues gap sizes and their frequency of occurrence increase, without end as pn → ∞. Proposition 2. As pn → ∞, within r 0 to r 0 2 the ai gaps increase infinitely in size and frequency. Proof. Because the residues exist as modular complement pairs, they have a mirror image symmetry distribution. Smaller residue gaps generally occur with much higher frequency, and large gaps systematically lower, among their total, and sub ranges. As pn increases, the residues become less dense and have more separation, and thus larger gaps, in higher frequencies, will be reflected within the primes r 0 to r 0 2 . As the range grows by p2 the number of primes grows : p2 /log(p2 ) and contain proportionally more k-tuples, which increase without end as pn → ∞. Fig 3. empirically shows the systematic increase in the size and frequency of the residue gaps for increasing Pn, required by (10) and (11). Fig 4. shows the slow initial, but then rapid, growth of the primes in r 0 to r 0 2 , while Fig 5. shows the steady growth of their k-tuples as pn increases. 8 Because coefficients a1 = a2 have a clear deterministic expression for all Pn, we can formulate a good estimate for prime gaps 2 and 4 (Twins|Cousins) for all Pn. We can simply say it’s the percentage of their gaps to its residue count times the number of primes from r 0 to r 0 2 , i.e. π(p, p2 ). For computational simplicity we can use π est (p, p2 ) = p · (p − 2)/2 · log(p), for a weaker estimate. T wins|Cousins count ≃ ⌈(a1 /rescntpn) · π(p, p2 )⌉ (12) 2 If we substitute the expressions for a1 , rescntpn, and π est (p, p ) we get: ⌉ ⌈∏ (pi − 2) p · (p − 2) · T wins|Cousins count ≃ ∏ (pi − 1) 2 · log(p) (13) To verify it works, let’s first use the parameters for P7, with r 0 = p = 11, rescntp7 = 48, and a1 = 15. The actual primes count π(11,121) = 26, thus: Twins|Cousins count ≃ ⌈15 · 26/48⌉ = 9. Using the weaker primes estimate of ⌈(11)(11 - 2) / 2·log(11)⌉, we get ⌈(15)(11)(9) / 96·log(11)⌉ = 7 Twins|Cousins primes. We see previously for P7 (and Fig 5.) the actual Twins|Cousins counts are 8|9 in the range 11 to 121, thus we get accurate estimates from both calculations. To test for a larger range, let’s use P97, whose r0 = p = 101. rescntp97 = a1|2 = ∏ ∏ (pi −1) = (2−1)·(3−1)·(5−1)·...·(97−1) = 277399690427737839953078806118400000 (podd − 2) = (3 − 2) · (5 − 2) · (7 − 2) · ... · (97 − 2) = 44148215542940151628274967912609375 π(101, 1012 ) = 1227 π est (101, 1012 ) = ⌈(101) · (99)/2 · log(101)⌉ = ⌈1083.3⌉ = 1084 Strong estimate: Twins|Cousins ≃ ⌈(a1 /rescntp97)·1227⌉ = ⌈195.3⌉ = 196 Weaker estimate: Twins|Cousins ≃ ⌈(a1 /rescntp97)·1084⌉ = ⌈172.5⌉ = 173 From Fig 5. we see the computed Twins|Cousins counts are 202|197 in the range 101 to 1012 . To establish with certainty an infinity of Twins|Cousins, et al, it’s only necessary to show at least one additional larger pair continually exists for some set of (not even all) Pn as pn → ∞. Here it’s established there is an estimable increasing large number of pairs for every Pn as pn → ∞. The computational forms for gap coefficients a3 to a34 (see Appendix) have also been determined, and reveal the structured deterministic relationship between all gap sizes. Each larger gap size frequency is a function of all smaller gaps. Thus their values can also be calculated for all Pn, and estimated within the range r 0 to r 0 2 for them. Once an ai comes into existence it can not then vanish (go to 0), or even decrease, as that would violate the constraints on the PGS gap structure. Thus we’ve established with certainty, prime gaps will always increase in frequency and size, precluding a last prime pair for any gap n as pn → ∞. Thus there are an infinity of all k-tuples. 9 Proof By Contradiction To say there are not an infinity of all k-tuples (i.e. a finite number) means empirically for all ai they become and remain zero (0) starting with some Pn. This mathematically requires the residues structure starting with this Pn to change in a mathematically permissible manner. Is this possible? 9 The structure of this proof is applicable for every gap coefficient ai , but I need only demonstrate it for a1 = a2 , as all other gaps are numerically related to them. Let’s imagine for some unknown Pn? with modulus pn? #, a1 = a2 reach some constant value, as pn increases. Under this scenario we know there still would be an infinity of Twins|Cousins, because all there needs to be at minimum is one additional larger pair continually found for just some Pn (let alone every Pn) as pn → ∞. Thus for there to be a finite number of Twins|Cousins, et al, we must have a1 = a2 = 0 starting with some Pn, and remaining so forever. But we know (see Appendix) that a3 is a function of a1|2 , a4 a function of a3 , etc, etc, thus it’s mathematically impermissible for this scenario to occur. It’s a mathematical absurdity for all the gap coefficients be zero, as there would be no residue gaps. Thus we have a clear contradiction. In addition, a1 = a2 conform to a deterministic relationship solely based on the modulus primes, and rapidly increase as pn → ∞. Thus a1 = a2 are never zero, and in fact increase within the range r 0 to r 0 2 for every Pn, precluding a last Twin|Cousin prime. To require an existing ai to permanently vanish creates a set of mathematically contradictory scenarios. For some Pn, its residues count would no longer be determined by the Euler Totient Function (so there are either more|fewer residues per modulus), and|or the residues are no longer modular complements (so their residues gaps distribution symmetry has changed). But the residue gaps cannot change without the residue values changing, which are the coprimes to modpn. Every conceivable scenario to establish a finite number for any gap size requires mathematical contradictions or absurdities. In fact, it’s easier to imagine by intuition alone there must be an infinity of k-tuples, than somehow mathematically envision and numerically establish their finality. Thus, to have a finite number for any prime gap requires its ai to become and remain zero, requiring a Pn’s structure to change in multiple impossible ways, which will affect every other gap. As there can be no finite number for any residue gap then consequently so too for any prime gap. 10 Predictive Results Ultimately, any proof must be able to explain known empirical results, and predict future ones. It’s shown we can compute a good minimum estimate for Twins|Cousins (and others) for any Pn. We can also establish when any residue gap first appears in some Pn, and then determine when it appears within the range r 0 to r 0 2 for some larger Pn. For example, a50 , which denotes residues gaps of 100, first occurs for P59 (because its PGS has on order 53 coefficients). Fig 5. shows a prime gap size of 100 first occurs for 503 < p < 1009. The exact value is p = 631; i.e. between 631 and 6312 the first prime pair of gap size 100 occurs among those 33,599 primes. Thus, while in general gaps of 100 start occurring between residues with P59, it takes until P619 to establish with certainty the first prime residue pair of this size, a span of 98 prime generators. While this simple process may not seem rapid, it is mathematically certain. The following list are the first prime pairs with the first multiple of 100 gaps sizes shown. • first instance of prime gap of 100 is (396,733; 396,833) • first instance of prime gap of 200 is (378,043,979; 378,044,179) • first instance of prime gap of 300 is (4,758,958,741; 4,758,959,041) • first instance of prime gap of 400 is (47,203,303,159; 47,203,303,559) • first instance of prime gap of 500 is (303,371,455,241; 303,371,455,741) 10 (It should be noted, the gaps don’t necessarily occur in linear order, as the first prime gap for 210, for the pair (20,831,323; 20,831,533), occurs well before the first prime pair gap 200.) Because their are an infinity of primes pn there are no theoretical upper bounds on this process. As the gap sizes increase their first, etc, prime residue pairs will become unimaginably large. But that’s OK. We need not determine their actual values, but merely establish with certainty (with this simple process) that they exist, and that there are an infinity of them of any size. 11 Conclusion The properties of Prime Generators allow for direct examination of the structure of the gaps between primes. They empirically show prime numbers, and their gaps, conform to a deterministic structure that determines their nature, numbers, and distribution. Residue gaps of any size n will first exist for some Pn, and occur in larger numbers for all larger generators. These residue gaps will ultimately appear and remain in the range r 0 to r 0 2 , becoming prime gaps for some Pn, and all larger. Thus, this simple process establishes the residue gaps only increase in size|frequency, and with ultimate certainty will appear as strictly prime gaps, whose k-tuples only increase without end as pn → ∞. References [1] The Use of Prime Generators to Implement Fast Twin Primes Sieve of Zakiya (SoZ), Applications to Number Theory, and Implications to the Riemann Hypothesis, Jabari Zakiya, 2019. - https://www.academia.edu/37952623The_Use_of_Prime_Generators_to_ Implement_Fast_Twin_Primes_Sieve_of_Zakiya_SoZ_Applications_to_Number_Theory_ and_Implications_for_the_Riemann_Hypotheses [2] The Segmented Sieve of Zakiya (SSoZ), Jabari Zakiya, 2014. - https://www.academia.edu/ 7583194/The_Segmented_Sieve_of_Zakiya_SSoZ [3] Twin Primes Segmented Sieve of Zakiya (SSoZ) Explained, Jabari Zakiya, J Curr Trends Comp Sci Res 2(2), 119-147, 2023. - https://www.opastpublishers.com/open-access-articles/ twin-primes-segmented-sieve-of-zakiya-ssoz-explained.pdf [4] k-tuples page - https://en.wikipedia.org/wiki/Prime_k-tuple [5] Twin Prime - https://en.wikipedia.org/wiki/Twin_prime [6] Polignac’s Conjecture - https://en.wikipedia.org/wiki/Polignac%27s_conjecture [7] PRIMES-UTILS HANDBOOK, Jabari Zakiya, 2016. - https://www.academia.edu/19786419/ PRIMES_UTILS_HANDBOOK [8] (Simplest) Proof of Twin Primes and Polignac’s Conjectures (video), Jabari Zakiya, 2021. https://www.youtube.com/watch?v=HCUiPknHtfY [9] Jumping Champions, Andrew Odlyzko, Micheal Rubinstein, Mark Wolf, Experimental Mathematics, 8(2), 107-118, 1999. - https://projecteuclid.org/journals/ experimental-mathematics/volume-8/issue-2/Jumping-champions/em/1047477055.full 11 Data The following data was derived using Ruby|Crystal scripts to generate and count the residue gaps. Listed here are all the residue gap coefficients ai for the first few prime generators. We observe: the sum of the columns for each Pn equals its residues count; the sum of the products of each ai by its gap size 2i equals modpn; and for each Pn there are on order pn−1 unique coefficients. Also for the Pn shown, the first instance for aprime (a3 , a5 , a7 , etc) equal 2. We also see the gaps frequency values oscillate up and down as they increase in size, with the smaller gaps numerically dominant in their frequency, and larger gaps initially occur with relatively much much lower frequency. This characteristic is a function of the computational forms of the ai , where each larger gap has a defined numerical relationship with the preceding smaller gaps and pn for its generator. Residue gap coefficients ai for all gaps 2i for given Pn Pn a1 · 2 a2 · 4 a3 · 6 a4 · 8 a5 · 10 a6 · 12 a7 · 14 a8 · 16 a9 · 18 a10 · 20 a11 · 22 a12 · 24 a13 · 26 a14 · 28 a15 · 30 a16 · 32 a17 · 34 a18 · 36 a19 · 38 a20 · 40 a21 · 42 a22 · 44 a23 · 46 a24 · 48 a25 · 50 a26 · 52 a27 · 54 a28 · 56 a29 · 58 2 1 3 1 1 5 3 3 2 7 15 15 14 2 2 11 135 135 142 28 30 8 2 13 1,485 1,485 1,690 394 438 188 58 12 8 0 2 17 22,275 22,275 26,630 6,812 7,734 4,096 1,406 432 376 24 78 20 2 19 378,675 378,675 470,630 128,810 148,530 90,124 33,206 12,372 12,424 1,440 2,622 1,136 142 72 20 0 2 Fig 3. 12 23 7,952,175 7,952,175 10,169,950 2,918,020 3,401,790 2,255,792 871,318 362,376 396,872 61,560 88,614 48,868 7,682 5,664 2,164 72 198 56 2 12 29 214,708,725 214,708,725 280,323,050 83,120,450 97,648,950 68,713,708 27,403,082 12,199,404 14,123,368 2,594,160 3,324,402 2,100,872 386,554 324,792 154,220 10,128 15,942 7,228 570 1,464 272 12 2 31 6,226,553,025 6,226,553,025 8,278,462,850 2,524,575,200 2,985,436,650 2,206,209,208 903,350,042 423,955,224 512,670,088 106,604,280 126,682,650 88,337,252 18,298,102 16,461,600 9,169,532 833,688 1,075,458 620,632 77,042 128,988 40,636 3,516 1,794 1,296 504 20 84 12 2 As new larger gaps appear within a PGS, it takes some time (i.e. some progression of generators) for them to appear within the range p to p2 of larger Pn where they become strictly prime gaps. The number of these residues constitute a dwindling percentage of the residue count for larger Pn, as shown below. This affects the rate of progression of Pn necessary to identify the strictly primes gaps. Pn residues count r0 to r0 2 count % of total residues 7 48 26 54.2 11 480 34 7.08 13 5,760 55 0.955 17 92,160 65 0.071 19 1,658,880 91 0.055 23 36,495,360 137 0.000375 29 1,021,870,080 152 0.0000149 Fig 4. Below shows the progression of gaps frequency within p to p2 for gap sizes shown, and the max gap. p max gap gaps of 2 gaps of 4 gaps of 6 gaps of 8 gaps of 10 gaps of 12 gaps of 14 gaps of 16 gaps of 18 gaps of 20 gaps of 22 gaps of 24 gaps of 26 gaps of 28 gaps of 30 gaps of 32 gaps of 34 gaps of 36 gaps of 38 gaps of 40 gaps of 42 gaps of 44 gaps of 46 gaps of 48 gaps of 50 11 8 8 9 7 1 Frequency of prime gaps (not complete) between p and p2 53 101 503 1,009 5,003 10,007 50,021 34 36 86 114 210 220 320 74 202 2,585 8,278 130,543 440,666 7,816,170 78 197 2,575 8,239 130,201 440,606 7,816,884 99 296 4,165 13,715 224,001 769,338 13,979,458 37 103 1,692 5,643 96,432 334,491 6,221,667 39 121 2,120 7,169 123,641 430,458 8,059,613 27 107 2,267 8,134 151,420 530,008 10,420,167 15 54 1,199 4,302 81,767 293,529 5,774,452 6 33 795 2,929 59,224 216,032 4,347,314 8 40 1,283 4,995 104,769 385,207 7,933,971 2 15 601 2,433 53,704 203,194 4,366,505 4 18 555 2,211 46,822 176,170 3,748,342 2 15 604 2,278 66,815 257,882 5,701,980 1 3 274 1,195 30,588 119,624 2,720,294 0 6 271 1,261 32,971 129,739 2,963,462 0 11 414 1,959 55,436 223,137 5,345,019 0 1 97 558 16,563 68,384 1,695,929 1 3 113 563 17,262 71,351 1,785,000 1 149 779 27,127 114,180 2,927,973 75 337 12,068 51,843 1,38,1811 90 436 14,320 60,853 1,640,477 83 486 19,568 86,754 2,438,771 23 205 7,745 34,939 1,001,765 24 158 6,514 29,372 866,337 29 203 10,790 49,904 1,501,630 16 110 5,803 27,544 857,165 Fig 5. 13 100,003 354 27,412,929 27,410,258 49,393,480 22,161,302 28,765,142 37,589,303 20,944,700 15,888,865 29,190,859 16,296,757 13,954,841 21,488,356 10,348,264 11,288,578 20,707,409 6,641,679 6,997,115 11,593,976 5,518,125 6,576,788 9,920,126 4,107,209 3,580,246 6,251,179 3,607,941 Here I use the data for p = 101 to visually show the oscillatory behavior of the gap sizes. We see from the data in Fig 5. this characteristic becomes more pronounced for larger p gap ranges. Larger ranges will have more local maxima|minima as they will generate more larger gaps. Each generator, thus, will have its own signature curve. We also see the local maxima|minima gap sizes exhibit an interesting characteristic: most of these ai indices are primes, i = 2, 3, 5, 11, 13, 17, or are powers of 2 or 3, i = 2, 4, 8, 9, 16, 27. It will be interesting to see the pattern for much larger gap sizes for increasing Pn. Prime gaps from p to p2 for p = 101 gaps 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 freq 202 197 296 103 121 107 54 33 40 15 18 15 3 6 11 1 3 1 Fig 6. We also clearly see the prominence of the smaller gaps and their expansion property. All the preexisting gaps are pushed toward the front for the first half mirrored gaps (as larger ones are included within the structure) and they will appear first, and in greater frequency than larger gaps, for each larger generator. But to be clear, we are observing the number of atomic gaps (between consecutive primes) not composite gaps (over multiple primes). The data shows, as expected, the ratio of Twins|Cousins is near unity (1) as their residue gaps are the same (providing the modular form framework to explain the Hardy-Littlewood Conjectures). We also see there will always be more Sexy Primes than Twins|Cousins, or any other individual k-tuple for the ranges shown. But with p = 503 we start to see gaps of multiples of 6 become the dominate local maxima of the gaps curves. In fact, the 1999 paper Jumping Champions, by Odlyzko, Rubinstein, and Wolf [9], suggests as we increase the number range, the most frequent prime gaps increase from 6, to 30, to 210, etc, i.e. are primorial gap sizes 3#, 5#, 7#, etc. 14 Here I show in more detail the slow growth rate of max gap sizes for increasing ranges p to p2 . p log10(p) max gap 11 1.0 8 19 1.25 14 p log10(p) max gap 10007 4.0 220 Max prime gap sizes from p to p2 31 59 101 179 317 563 1,009 1.5 1.75 2.0 2.25 2.5 2.75 3.0 20 34 36 72 72 86 114 17783 4.25 248 31627 4.5 282 56237 4.75 320 100003 5.0 354 177823 5.25 456 1,783 3.25 148 316233 5.5 464 3,163 3.5 154 526337 5.75 486 5,623 3.75 210 1000003 6.0 540 Fig 7. This graph quantifies the slow expansion. As p increases orders of magnitude its PGS max gap grows much slower. For p of order 103 the max gap reaches 102 , but only increases to 5 · 102 for p of order 106 . We can create growth curves for all the other gap sizes to see their growth rate. It should be noted again, though while this graph is technically accurate, it doesn’t tell the whole story, as the gaps don’t always occur in linear order. For example, the first prime gaps for 210, 220, 248, etc, occur for prime values much smaller than for the first prime pair with gap 200. Also, primes gaps seem to occur in clusters. Primes with (relatively) small gaps seem to cluster in progression. As we journey higher into the number space we start to observe more and larger prime gaps (in fact an infinity of them), regions I call prime vacuums (or deserts). The smaller gap clusters exist around the vacuums, which using classical numerical techniques makes searching for extremely large Twins, Merseene Primes, etc harder. You ideally want to be able to identify where the vacuums are and avoid them. We can use the residue gaps profiles for PGS to confine searches accordingly based on the goals. See [1]. Thus the data illustrate the distribution of primes is not random, but in fact deterministic, and conform to the described properties manifested within the structure of prime generators. 15 Appendix Infinite Progression of Primes From the Prime Number Theorem (PNT) (https://en.wikipedia.org/wiki/Prime_number_theorem) it has been proved the number of primes up to any value x is on order x/log(x), or better Li (x) (log integral x). I use equation (8) (for computational simplicity) x/log(x) to estimate the number of primes between any random prime p (or really any value x ) and p2 , per the PNT. The Pn residues are the integers pn < ri < modpn coprime to modpn. The Euler Totient Function (ETF) tells us their exact number. Thus it’s clear, the {ri } must include all the primes, and their coprime multiples < modpn, necessary to satisfy the ETF residues count. Each Pn eliminates all its modulus primes multiples from consideration. Since the first residue r 0 of every Pn is the next prime > pn , its first multiple in its residue set (pcs table) is the multiple with itself, i.e. r 0 2 . Therefore, the residues between r 0 to r 0 2 can only be the consecutive primes in that interval, as they are not multiples (the only non-multiples) of the modulus primes < r 0 2 . 2 2 2 And the PNT estimates their numbers are of order p /log(p ) - p/log(p), or better Li (p ) - Li (p). However, for each specific generator Pn we can compute easier a simpler estimate. We know the number of modulus primes for any Pn, I’ll note as π(modpn). Thus the primes < r 0 2 , for r 0 = p 2 2 are: p /log(p ) - π(modpn). For the previous example for P23, with r 0 = 29, a simpler calculation 2 2 is then: ⌈(841)/log(841) - 9⌉ = ⌈115.87⌉ = 116 as before. In fact, we can just use p /log(p ), here ⌈841/log(841)⌉ = ⌈124.88⌉ = 125, as π(modpn) is relatively so much smaller as p2 becomes larger. Thus, since we know each generator Pn always generates the consecutive primes r 0 to r 0 2 , we can use these primes to construct a larger Pn, and keep repeating this process as many times as we want to generate as many consecutive primes groups we want, and thus can also then observe, record, and count, the exact gap structure of all the primes, into infinity. The graph below shows the growth in the number of new primes in r 0 to r 0 2 for each of the first 100 primorial Pn generators. We see it has the classic x2 parabolic curve, as the number of primes will grow without end as more primes are used to construct larger primorial generators. Fig 8. 16 Here we see the ratio of the number of new primes to primorial primes. It has a much more linear profile, as their growth appears fairly constant for the first 100 primorials. It’ll be interesting to see if it approaches some asymptotic limit as the primorial primes increase by orders of magnitude. Fig 9. Modular Complement Property Using clock math, we see residues exist as modular complement pairs, and prime generator sequences have mirror image symmetry , as a direct property of their modular forms. Any even n can be the modulus for a cyclic integer generator (a ring Zn) we can visualize as a clock of n hours. A 12 hour clock has a modulus of 12 with mod values 0 – 11. We see if we draw horizontal lines between the hours left-to-right, their sums equals 12, and also see this if we fold the clock on its vertical axis. Moduli with multiple factors of 2 (as here) have even midpoint|bottom values, thus the bottom gap is 2. For primorial moduli, et al, with one factor of 2, the midpoint is odd, and the bottom (pivot) gap is 4 between the odd values on each side. The top gap is 2, so primorials have equal odd numbers of gaps of 2 and 4, while all others occur evenly on the clock. When we form the prime generator P12, for mod12 we only use the residues coprime to 12, i.e. {1, 5, 7, 11}, where (1, 11) and (5, 7) are modular complement pairs. Eliminating the non-coprime values creates the P12 generator with these 4 residues, and its mirror image gap distribution. Any even n > 2 will have a modular form with these modular complement properties, for every Pn. Fig 10. 17 Reduced Primorials The principal, and reduced primorials of rank r, play a fundamental role in∏the construction of the n ai residue gap coefficients values. They have form: (pn − r)# = p−r n # = pi ≥r (pi − r),where for pi = r, (pi − r)# = 0#=1, similar to 0!= 1. Below is a table of the reduced primorials for the first 10 primorials. n Pn r=0 r=1 r=2 r=3 r=4 r=5 r=6 r=7 r=8 r=9 r=10 r=11 r=12 r=13 r=14 r=15 r=16 r=17 r=18 r=19 r=20 r=21 r=22 r=23 r=24 r=25 r=26 r=27 r=28 r=29 1 2 2 1 1 2 3 6 2 1 1 3 5 30 8 3 2 1 1 4 7 210 48 15 8 3 2 1 1 Reduced Primorial Values Pn−r # 5 6 7 8 11 13 17 19 2,310 30,030 510,510 9,699,690 480 5,760 92,160 1,658,880 135 1,485 22,275 378,675 64 640 8,960 143,360 21 189 2,457 36,855 12 96 1,152 16,128 5 35 385 5,005 4 24 240 2,880 3 15 135 1,485 2 8 64 640 1 3 21 189 1 2 12 96 1 5 35 1 4 24 3 15 2 8 1 3 1 2 1 1 Fig 11. 18 9 23 223,092,870 36,495,360 7,952,175 2,867,200 700,245 290,304 85,085 46,080 22,275 8,960 2,457 1,152 385 240 135 64 21 12 5 4 3 2 1 1 10 29 6,469,693,230 1,021,870,080 214,708,725 74,547,200 17,506,125 6,967,296 1,956,955 1,013,760 467,775 179,200 46,683 20,736 6,545 3,840 2,025 896 273 144 55 40 27 16 7 6 5 4 3 2 1 1 Gap Coefficients ∏ It was previously established: a1 = a2 = (pn −2)# = (podd −2) = (3−2)·(5−2)·(7−2)·...·(pn −2). I have also determined the recursive forms for a1 - a7 . For any generator Pn, with last modulus prime pn , its gap coefficients ai are a function of pn and the preceding generator coefficients a′i . a1 a2 a3 a4 a5 a6 a7 = = = = = = = a′1 · (pn − 2) a′2 · (pn − 2) a′3 · (pn − 3) a′4 · (pn − 4) a′5 · (pn − 5) a′6 · (pn − 5) a′7 · (pn − 7) + + + + + a′2 + a′1 a′3 a′4 · 2 + a′3 a′5 · 6 − a′4 · 2 a′6 · 3 − a′5 · 3 + a′4 · 4 The P37 gap coefficients distribution has now also been directly generated, a1 = 217,929,355,875 a2 = 217,929,355,875 a3 = 293,920,842,950 a5 = 108,861,586,050 a6 = 83,462,164,156 a7 = 34,861,119,734 a9 = 21,218,333,416 a10 = 4,814,320,320 a11 = 5,454,179,550 a13 = 918,069,454 a14 = 857,901,000 a15 = 535,673,924 a17 = 69,404,898 a18 = 46,346,428 a19 = 7,381,190 a21 = 4,153,336 a22 = 526,596 a23 = 291,342 a25 = 91,392 a26 = 8,912 a27 = 25,320 a29 = 1,654 a30 = 452 a31 = 26 a33 = 24 and is shown below. a4 = 91,589,444,450 a8 = 16,996,070,868 a12 = 4,073,954,144 a16 = 58,664,256 a20 = 10,176,048 a24 = 239,760 a28 = 2,952 a32 = 48 We can now calculate the gap estimates within the range p to p2 for a1 - a7 . Comparing data from Fig 5. let’s calculate the estimates for a1 - a7 for p to p2 for r 0 = p = 53. This means we have to find all those coefficients values up to P47. Below are their calculated values starting from P37. pn a1 a2 a3 a4 a5 a6 a7 Calculated residue gap coefficients ai for gaps 2i for given Pn 41 43 47 8,499,244,879,125 348,469,040,044,125 15,681,106,801,985,625 8,499,244,879,125 348,469,040,044,125 15,681,106,801,985,625 11,604,850,743,850 481,192,519,512,250 21,869,408,938,627,250 3,682,730,287,600 155,231,331,960,250 7,156,139,793,803,000 4,396,116,829,650 186,022,750,845,750 8,604,610,718,954,250 3,474,628,537,016 151,047,124,809,308 7,149,653,083,144,936 1,475,437,583,074 65,082,209,263,162 3,119,286,820,258,154 Fig 12. We can now use P47’s calculated ai to find their range estimates: gapsi ≃ ⌈ai ·π(p, p2 )/rescntp47⌉ For p = 53, π(p, p2 ) = 394, and rescntp47 = gaps 53 to 532 actual estimated a1 |a2 74|78 73 ∏p47 p2 a3 99 102 Fig 13. 19 (pn − 1) = 85287729364992000, gives values: a4 37 34 a5 39 40 a6 27 34 a7 15 15 There’s also an algebraic way to generate the ai values without recursion, using reduced primorials. The table below shows the first 20 ci,j rational coefficients, which when multiplied by the respective reduced primorial Pn−r # column values, and summed across each row, will compute the ai resdue gap values for any Pn generator. Pn−r # c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 r=2 1 1 2 1 4 3 2 6 5 1 2 4 3 10 9 2 12 11 6 5 8 3 1 16 15 2 18 17 4 3 r=3 -2 -2 -3 -7 -5 -5 −23 2 −39 4 −63 8 -17 −209 20 −185 16 −1207 40 −127 10 −765 56 −15543 560 −2499 160 −1139 56 r=4 1 2 10 28 3 12 100 3 116 3 632 21 1738 21 11090 189 456 7 12986 63 18833 189 1406 13 592204 2457 1777744 12285 7658 39 ai (n) = ci,j Pn−r # r=5 r=6 r=7 -2 -3 -6 -22 -40 −175 6 −344 3 −1563 16 −325 3 −5221 12 −2999 12 −6565 24 −391649 576 −48925 112 −89845 144 1 6 24 72 5 r=9 r=10 r=11 28 5 16 3 482 5 2287 15 4424 27 94618 135 262864 495 135476 135 −45 4 −83 2 −675 16 −4151 16 −60093 320 −1745 4 20 3 128 21 1516 21 340 9 2848 21 −14 3 -2 108 -21 4224 35 662 5 25376 35 3651 7 220944 385 125544 77 801716 715 658414 385 −119 3 ∑k r=8 -42 −4285 12 −2107 6 −15411 40 −155981 120 −684359 720 −94969 60 -8 −(1+j) We directly compute each ai (n) value as: ai (n) = j=1 ci,j Pn #, for their k row values. Thus as before: a1 (n) = a2 (n) = c1|2,1 Pn−2 # = (1)(pn − 2)#, are the 2nd reduced primorials. For a computaionally longer example, let’s compute a8 (residue gaps of 16) for P23, i.e.P9 . To compute it for any Pn we have: a8 (n) = c8,1 Pn−2 # + c8,2 Pn−3 # + c8,3 Pn−4 # + c8,4 Pn−5 # + c8,5 Pn−6 # For P23, p9 = 23 (for the 9th primorial) and we use the P9−r # reduced primorial table values. a8 (9) = (1)P9−2 # + (−5)P9−3 # + (12)P9−4 # + (−6)P9−5 # + (1)P9−6 # = (7952175) − 5(2867200) + 12(700245) − 6(290304) + (85085) = 362376 We see this matches the value in Fig 3., which were obtained by brute force computation. At time of writing, the values up to c80 have been determined, to compute a1 − a80 for any Pn . However, a presentation of their derivation is beyond the scope of this paper. 20 Numerical Gap Derivations The ai coefficients can be numerically determined by the constrained system of equations for Pn: ∑ modpn = pn # = ai · 2i = 2 · a1 + 4 · a2 + 6 · a3 + ... + 2n · an (14) rescntpn = (pn − 1)# = ∑ ai = a1 + a2 + a3 + ... + an (15) As pn # is an even value c1 and (pn − 1)# an even value c2 we can reduce the equations to: c1 /2 = a1 + 2 · a2 + 3 · a3 + ... + n · an (16) c2 = a1 + a2 + a3 + ... + an Oddness of a1 and a2 For P2 we only need to use: 2# = 2 = 2a1 (17) This numerically establishes a1 = 1 for P2 as the single (odd) value for gap size 2 for its PGS. For P3 we have c1 = 3# = 6 and c2 = (2 − 1) · (3 − 1) = 2, and we are constrained to only having the two nonzero coefficients a1 and a2 , which gives: 3 = a1 + 2a2 (18) 2 = a1 + a2 The only solution is a1 = a2 = 1, matching the known odd occurrences for gaps 2 and 4 for P3. For P5 we have c1 = 5# = 30 and c2 = 8, and are constrained to only having the nonzero coefficients a1 , a2 , and a3 which gives: 15 = a1 + 2a2 + 3a3 (19) 8 = a1 + a2 + a3 We now create the system of equations: 2R2 - R1 and 3R2 - R1, 1 = a1 − a3 (20) 9 = 2a1 + a2 which after rearranging gives: a3 = a1 − 1 (21) a2 = 9 − 2a1 We solve this by picking the value for a1 that produces a2 and a3 that satisfy equations (19). Notice a2 is odd for any value of a1 , and because we know a3 is even a1 must be odd, and constrained to 1 or 3 (5 makes a2 negative). Only a1 = 3 works, producing a2 = 3 and a3 = 2, the known PGS values for P5. Again we see, now purely through numerical methods, that a1 = a2 and numerically required to be odd, which matches the computational form for these Pn: a1|2 = a′1|2 · (pn − 2). 21 Let’s continue for P7, with c1 = 7# = 210 and c2 = ∏p7 p2 (pn − 1) = 48. 105 = a1 + 2a2 + 3a3 + 4a4 + 5a5 (22) 48 = a1 + a2 + a3 + a4 + a5 Now do R1 - R2, to eliminate a1 , and R1 - 2R2, to eliminate a2 , and after rearranging gives: a2 = 57 − 2a3 − 3a4 − 4a5 (23) a1 = −9 + a3 + 2a4 + 3a5 Again, a1 and a2 are odd as a3|4|5 are even (due to their mirror symmetry). This problem is solvable using linear programming algorithms e.g. the Simplex Method. It can be characterized using their prime generators properties to produce the ai values for P7, i.e. a1 = a2 = 15, a3 = 14, a4 = 2, and a5 = 2.1 For all larger Pn, a1 |a2 will have similar forms as (23) with more ai terms. Solving for larger ai However, we really want a system of equations where the larger gap coefficients are functions of the smaller ones, to reflect the order of their relational structure we see empirically expressed in their computational forms. Thus, because we know a1 = a2 we can transform (22) to: (105 − 3a1 ) = c3 = 3a3 + 4a4 + 5a5 (24) (48 − 2a1 ) = c4 = a3 + a4 + a5 We now create a new system, solving for a5 , and performing 5R2 − R1 and solving for a4 : a5 = c4 − a4 − a3 (25) a4 = 5c4 − c3 − 2a3 We can now pick a3 to determine a4 , and then a5 , which gives us all the ai . For P7, a1 = a2 = 15 gives c3 = 60 and c4 = 18 creates: a5 = 18 − a4 − a3 (26) a4 = 30 − 2a3 Because a3|4|5 are > 0 and even, requires 2 ≤ a3|even ≤ 14, the only solution is, again, a3 = 14, a4 = 2, and a5 = 2. Creating the equations in this order provides for computation of the lower values for larger gaps. As the gaps become larger we’ll see more of the oscillating nature of their values as functions of smaller gaps, as shown in Fig 6. Thus we illustrate again using numerical methods, the properties of prime generators determine the unique solution to the system of constraints for each Pn, which show the gap coefficients ai will only increase in frequency value for all gap sizes, as the Pn moduli pn # increase as pn → ∞. 1 Using Simplex Calculator at http://cbom.atozmath.com/CBOM/Simplex.aspx?q=is with following constraints, produces known ai values: MIN Z = x1 + 2x2 + 3x3 + 4x4 + 5x5 subject to: x1 + 2x2 + 3x3 + 4x4 + 5x5 = 105; x1 + x2 + x3 + x4 + x5 = 48; x1 <= 15; x2 <= 15; x3 <= 17; x4 <= 13; x5 <= 10; x1 >= 3; x2 >= 3; x3 >= 2; x4 >= 2; x5 >= 2; and x1 , x2 , x3 , x4 , x5 >= 0. 22 Closing Thoughts Since the 2013 release of Yitang Zhang’s paper2 that for some integer N < 70 million there are infinitely many pairs of primes that differ by N, there has been a fury of activity to reduce its bound to a smallest gap size. Included now is the quest to solve problems regarding questions of small and large gaps.3 The work presented here proposes to establish with certainty there are an infinity of prime pairs that differ by any gap size, large and small. Using strictly numerical approaches will likely continue to be fruitless to definitively answer questions about prime gaps. If you want to understand and characterize the nature of prime gaps the most direct (and easiest) approach is to strictly work within the domain of prime gaps. Prime Generator Theory (PGT) provides the theoretical, philosophical, and numerical framework to do this, which current analytical and numerical methods alone are not equipped to do. At the beginning of the 20th Century, Relativity Theory was imagined by Einstein to provide both a qualitative and quantitative framework to better understand and explain how nature works. Initially it was resisted, but ultimately was (had to be) embraced because it worked. It could quantitatively answer questions about the known behavior of nature other theories couldn’t, and accurately predict and explain previously uncontemplated behavior. And continual experimental testing has reaffirmed its validity (for the reality we are aware of), over and over. Here at the start of the 21st Century, I believe PGT shares a similar role in the field of math. It provides a better framework to qualitatively and quantitatively understand, characterize, explain, and predict the behavior of primes. Resistance has run mostly along the lines of questioning language, the meaning of terminology, being too simplistic, the perceived lack of rigor, etc. These are complaints more about its qualitative nature, and|or epistemological basis for knowing, than a refutation of its theoretical foundations or its empirical results and predictions. The content herein is a major revision of the earlier versions, to present its findings in a clearer and more “mathematician friendly” format, and to present new information and findings. I would ask whatever it may seem to lack in traditional mathematical rigor not be a deterrent from recognition of its mathematically sound theoretical under girding. Judge it on the merits of the evidence of its findings and results, which I contend overwhelmingly establish with certainty it claims. Undoubtedly the work presented here touches just the surface of a body of knowledge begging to be explored and revealed. Hopefully the curious will take up the challenge to do just that, and share their findings, and apply them to the myriad of known problems waiting to be solved, while contemplating and proposing new ones heretofore unimagined. 2 Bounded 3 Small gaps between primes; https://annals.math.princeton.edu/2014/179-3/p07 and Large Gaps Between the Primes; https://www.youtube.com/watch?reload=9&v=pp06oGD4m00&t=425 23