On the Infinitude of Twin Primes: A Modular Proof
The term ”twin prime” comes from the German mathematician Paul Stäckel, who first coined the term back in the late 19th century. In the middle of that same century, back in 1849, the French mathematician Alphonse de Polignac put forward what is now called the Twin Prime Conjecture. It states that:
There are infinitely many pairs of prime numbers, (p) and (p + 2), which differ by exactly 2.
This conjecture remains an open question. As is the question of whether or not there is an infinite number of twin primes. They are but two of the many unresolved questions about the distribution of prime numbers.
Now, there are certain proven theorems about the distribution of prime numbers, such as Bertrand’s Postulate, which states that:
There is always a prime number located between: (n < prime < 2n).
And there are other conjectures, ones that most mathematicians believe are true but remain unproven, such as Legendre’s Conjecture, which posits that for two squared numbers:
There is always a prime number located between: n² < prime < (n + 1)².
Thanks to an observation of the modular nature of primes and their multiples, we put forward a new conjecture that there is a similar dynamic for twin primes, such that:
Conjecture: There is always a pair of twin primes located between:
p² < Twin Prime < (p²+ 4p).
Or, if you prefer a more elegant mathematical formulation, one that features twin primes in the expression:
Conjecture: There is always a pair of twin primes located between:
p² < Twin Prime < (p + 2)² — 4.
These are equivalent expressions and give the same result.
On The Infinity of Primes and Twin Primes
As Euclid proved back in ancient times, there is an infinitude of primes. Over the centuries, many mathematicians have retraced his steps and proven this to be true. They’ve done so in novel ways, confirming there are, indeed, prime numbers extending up to infinity.
Most recently, in 2014, great progress was made in our quest to understand the nature of twin primes. It was sparked by a paper by Yitang Zhang on the bounds for prime gaps. His landmark work was soon followed by subsequent groundbreaking work by a pair of Fields Medal winners, James Maynard and Terrence Tao.
It seemed a proof of the Twin Prime Conjecture dangled just within reach.
Perhaps very soon we’d finally be able to prove whether there are infinite twin primes, or not. Except the methods they used to make such swift progress, the techniques which enabled their tremendous leaps, have now bumped up against their natural mathematical limits. And thus, a new path forward must be attempted.
In our concerted efforts to understand prime numbers, perhaps by fixing our focus so tightly on the prime numbers and only the prime numbers, by relying on estimation tricks of the trade such as Big O notation, and by basing our efforts on other estimates such as the prime counting function, mathematicians have lost sight of the natural environment of the prime numbers: the number line.
In our pursuit of the novel, the abstract and theoretical, as well as the algorithmic, perhaps we’ve overlooked a rather simple method that reveals the seemingly random and unpredictable nature of prime numbers. Perhaps, we’ve overlooked it precisely because we ignored the very nature of those numbers we call natural numbers. Instead, by refocusing our attention on the environment of the prime numbers, by stalking them where they live on the number line, that is how we discovered this new conjecture for twin primes. We found it by contemplating their absence.
In other words, we sought out the places on the number line where we know that integers are never prime numbers. And that turned out to be the secret to finding twin primes
Lemma 1: All prime numbers > 3 are of the form: +/– 1 (mod6).
Lemma 2: Not all numbers of the form: +/–1 (mod6) are prime numbers, such as squared prime numbers, as well as the products of two (or more) prime numbers.
Lemma 3: When two (or more) prime numbers are multiplied together they form a prime composite also of the form +/– 1 (mod6).
Lemma 4: All numbers of the form: +/– 1 (mod6) form a closed set for multiplication.
Based on the above lemmas, it follows that numbers of the form +/– 1 (mod6) form a closed set under multiplication, and that such numbers of the form (6k +/– 1) are either a prime number, or a product of at least two primes, which can be expressed as:
Prime numbers: (6k — 1) and (6k + 1)
Prime number squared: (6k — 1)(6k — 1) or (6k + 1)(6k +1)
Prime composites: (6k — 1)(6k + 1) or (6k — 1)(6k — 1) or (6k +1)(6k + 1)
The relationship between prime numbers, squared prime numbers, and prime composites is what we will leverage to prove Polignac’s version of the Twin Prime Conjecture, and to examine and prove our new conjecture about the dynamic of twin primes:
The (Mod6) Framework and the Sieve of Eratosthenes
In order to familiarize ourselves with the +/– 1 (mod6) number line for prime numbers and their composites, which we’ll be exploring, let’s begin with the Sieve of Eratosthenes.
When we apply it to the number line, the Sieve of Eratosthenes recommends that we select the numbers 2 and 3, and remove all of their subsequent multiples. With the exile of the multiples of 2, we lose all the even numbers from the number line. Next, we jettison all the non-even multiples of 3, such as: 9, 15, 21, 27,… and so on.
After removing the even multiples of 2, and odd multiples of 3, all that remains on the number line are integers of the form: +/– 1 (mod6):
Those numbers are all of the form (6k +/– 1), or +/– 1 (mod6), and they will either be a prime number, a squared prime number, or a prime composite, which is a product of two or more prime numbers. Squared prime numbers are always of the form: (6k + 1). Whereas, primes and prime composites can take the form of either: (6k + 1) or (6k — 1).
Next, we’ll rearrange this number line so that the Sieve of Eratosthenes reflects this same modular form. Which means, we’ll double up the numbers of the form (6k +/– 1). One group above and one below the number line:
This modular arrangement of the number line helps to visualize the distribution of primes and their multiples, the prime composites.
As we’ll show, each prime number and its multiples follows a unique pattern. It’s a semi-arithmetic progression. One that’s determined by a prime’s unique value, multiplied by six.
This (+6p) rate we’ll call a prime’s “wavelength.”
Lemma 5: For every prime (p) there is a unique “wavelength” equal to (6p).
For example, (5)(6) = 30, and (7)(6) = 42.
There are two ”waves” of arithmetic progression we can use to track the multiples of a prime. If we add the value (+6p) as the common difference between terms, the result is an arithmetic sequence. But in this case, we prefer to think of the progression as a wave, for reasons that will soon be self- evident.
We need two “waves” for each prime number because if we start our count of the arithmetic progression at (+p), we will only “count” half of the possible prime composites. Instead we must start our “count” on both sides of the zero. One “wave” starts at ( — p), while the other starts at (+ p).
Lemma 6: There are two (+6p) waves of arithmetic progression that track the multiples of a prime.
For example, if (p = 5), the “wave” value would be: (5)(6) = 30. Which gives the value for its arithmetic progression: (+ 30). If we add that value as the common difference of the sequence, starting at (–5), the result is:
When we start the sequence at (p = –5), the resulting “wave” alternates between multiples of the form (5)(6k — 1):
(–5) — — — — — — — — (55) — — — — — — — — — (115) — — — — –––→
–∥ — — — — ∥ — — — — — ∥ — — — — — ∥ — — — — — ∥ — — — — — ∥ — — →
— — — — –(25) — — — — — — — — — (85) — — — — — — — — —(145)–→
Additionally, there’s the positive “wave” for (p = +5), which “hits” all the
values for (5)(6k + 1), due to the same (+30) arithmetic progression:
— — — — –(35) — — — — — — — — — (95) — — — — — — — — — (155)–→
–∥ — — — — ∥ — — — — — ∥ — — — — — ∥ — — — — — ∥ — — — — — ∥ — — →
(+5) — — — — — — — — (65) — — — — — — — — —(125) — — — — —–→
Which has the same common difference for its arithmetic sequence.
We can combine these two wave-like patterns and track both at the same time on our modified number line. If we start our “count” at the positive value (+5), the new “singular wave” will follow that same arithmetic progression (+30), hitting all the values for (5)(6k +/– 1).
However, our newly-combined and now “singular wave” no longer has a common difference between terms. Instead, the pattern oscillates with two different values: (+4p) and (+2p).
One increase is: +2/3(6p). While the other increase is: +1/3(6p).
The now “singular wave” propagates at the rate: (+4p, +2p, +4p, +2p, +4p…)
As it progresses, unevenly, it never misses a multiple of: (5)(6k +/– 1).
This same dynamic holds true for any prime (p), and its multiples, the prime composites of the form: (6k +/– 1).
It’s important to note that this offset progression of our newly-combined “singular wave” means it’s no longer a true arithmetic progression. The difference between terms must be the same for it to qualify as such. And since the common difference is no longer the same, the sequence is no longer an arithmetic progression. At best, it could be called a semi-regular arithmetic progression.
Yet, as we’ve shown above, when combined together, our new “singular wave” dynamic snakes through all the values (mod6) of our modified Sieve of Eratosthenes number line. Which means it “hits” all of the multiples of any prime (p).
For example, if (p = 5), the semi-arithmetic progression is:
(5 + 4p) = 25; (25 + 2p) = 35; (35 + 4p) = 55; (55 + 2p) = 65…
If (p = 7), then that same (+4p, +2p…) semi-arithmetic progression is:
(7 + 4p) = 35; (35 + 2p) = 49; (49 + 4p) = 77; (77 + 2p) = 91…
If we add two alternating rates of increase, then the difference between terms qualifies as a true arithmetic progression.
For example, (7 + 4p) = 35; (35 + 2p) = 49; and (35 + 2p) = 49; (49 + 4p) = 77 and thus, (77–35) = (49–7) = 42 = (+6p)
To track the semi-arithmetic progressions for a prime (p), but do so in terms of (mod6) units of increase, we can calculate the unique values (x, y), for each prime number.
To determine the values for (x, y), we calculate:
Let x = p − ((p + / − 1) ÷ 3))
And let y = ((p + / − 1) ÷ 3))
Take for example, the case (p = 7), the semi-arithmetic progression in terms (mod6) would be: (+5, +2, +5, +2, +5, +2…).
In terms of (mod6), the (x, y) patterns of semi-arithmetic progression for the first few primes are:
5: (+3, +2, +3, +2, +3, +2…)
7: (+5, +2, +5, +2, +5, +2…)
11: (+7, +4, +7, +4, +7, +4…)
13: (+9, +4, +9, +4, +9, +4…)
17: (+11, +6, +11, +6, +11, +6…)
19: (+13, +6, +13, +6, +13, +6…)
23: (+15, +8, +15, +8, +15, +8…)
A Pascal-Like Triangle for the Arithmetic Progressions of Primes and their Multiples
To help further visualize the patterns of the semi-arithmetic progressions of primes and their multiples, we can create a Pascal Triangle-like arrangement, in terms of values (mod6).
The red numbers are the (mod6) values of the squares of (5, 7, 11 and 13). The diagonals are the intersections for the (6k +/– 1) values. The resulting pyramid pattern depends on the prime’s unique (x, y) values.
As we see below, if we add-in one more term to the pyramid, as the diagonals descend from right to left they track the addition of the alternating values (x, y), such that:
Descending from right to left, the diagonals start with the value (k) for the prime numbers, in terms of:(6k +/ — 1), and then the pattern of the semi-arithmetic progression alternates with the addition of the (x, y) values.
For example, for (p = 5), the value for (k) = 1, and so:
(1 + 3) = 4; (4 + 2) = 6; (6 + 3) = 9; (9 + 2) = 11
And for (p = 7), the value of (k) is also 1, but the pattern is different, due to its unique values for (x, y):
(1 + 5) = 6; (6 + 2) = 8; (8 + 5) = 13; (13 + 2) = 15
For (p = 11) and (p = 13), they share the same value for (k), and their unique
progressions are:
(2 + 7) = 9; (9 + 4) = 13; (13 + 7) = 20; (20 + 4) = 24
(2 + 9) = 11; (11 + 4) = 15; (15 + 9) = 24; (24 + 4) = 28
As we saw in the pyramids above, certain values are “skipped over.” This is due to the unique (x, y) values of the prime numbers. And this same ”skipped over” dynamic will continue to occur as the pyramid is enlarged.
Or in other words, as we count up to infinity.
The Uneven Arithmetic Progressions of the Twin Prime Conjecture Dynamic
The key dynamic for our modified Twin Prime Conjecture is how the multiples of the prime numbers occur with a predictable regularity. This is due to the fact they are all of the form: (6k +/– 1).
As well, as we’ve seen, the unique semi-arithmetic progressions of the primes and their multiples reveals a pattern wherein certain values (mod6) are always “skipped over” by the subsequent multiples of the primes.
Now, let’s examine the (+4p) range to determine how the ”waves” of the prime numbers and their prime composites and the ”skipped over” values, all relate to the distribution of twin primes, according to our Twin Prime conjecture:
p² < TwinP rimes < (p² + 4p)
The difference between (p²) and (p² + 4p) is obviously (+4p). Which, as we’ve shown earlier, is one of the alternating rates for the semi-arithmetic progressions of prime numbers. To examine this dynamic, we first need to count the number of (mod6) values of the (+4p) extension. That calculates the number of locations where we’ll find numbers of the form (6k — 1) and (6k + 1). In other words, in terms (mod6), the number of spots on the number line that might be primes or prime composites in the (+ 4p) range.
It’s a simple operation, we divide: (4p/6).
Which, obviously, can be rewritten as: (2p/3)
For example, let (p = 19), the extended (+ 4p) range has the value: 4(19) =76.
And we calculate: (76/6) = (38/3) = 12.6666
This tells us there are 12 values on the (mod6) number line in the (+ 4p) range.
This also tells us the potential twin prime pairs, in terms of (mod6), and that they are located between the values: (61–73).
In natural numbers, the 12 pairs of values are: (365, 367); (371, 373); (377, 379); (383, 385); (389, 391); (395, 397); (401, 403); (407, 409); (413, 415); (419, 421); (425, 427), (431, 433).
Next, we determine the lesser primes whose “wavelengths” will intersect with those 12 sets of paired numbers.
To determine that, we calculate: √(p² + 4p).
The result tells us both the limit and the number of primes < (p² + 4p) that can possibly intersect.
For example, if (p = 19), the upper limit is 17.
That’s because 19 is a number of the form (6k +1), and thus, the first step of its arithmetic progression after it’s squared is (+4p). In other words, as we’ve shown earlier, we do not need to include 19 in our count of the lesser primes, since (+4p) is the next multiple after (p²), which means 19 won’t “hit” any of those 12 possible twin prime spots on the number line.
If (p = 19), the lesser primes that intersect in the range (+4p) are: (5, 7, 11, 13, 17). Which also tells us the number of the lesser primes is: 5.
Next, we calculate the number of times their unique (+6p) “wavelengths” intersect with one of the 12 (mod6) values in the (+4p) range:
5: 5 times (365, 385, 395, 415, 425);
7: 4 times (371, 385, 413, 427);
11: 2 times (385, 407);
13: 2 times (377, 403);
17: 2 times (391, 425);
for a grand total: 15 times.
Note that a few of those numbers share the same value for (k). To eliminate a possible twin prime pair from our list of the 12 potential pairs, we only require one value for (6k +/– 1), to be “hit.” Or in other words, we only need to intersect with one (+/–) value for (k). The “waves” of the prime multiples can “hit” either (6k — 1) or (6k + 1).
Thus, we can remove any redundancies from our total:
5: 5 unique hits (365, 385, 395, 415, 425);
7: 1 unique (371);
11: 1 unique (407);
13: 2 unique (377, 403);
17: 1 unique (391);
for a grand total: 10 unique “hits”
We’re left with the 10 “unique” intersections of the “wavelengths” for the multiples for the lesser primes.
If we subtract those 10 unique “hits” from the 12 potential twin prime pairs in the (+ 4p) range, we find there are 2 values (mod6) where neither value, (6k — 1) or (6k + 1), is intersected by a “wavelength” of a lesser prime. In other words, there are two values for (k) which are free from any intersection of a multiple of the lesser primes. Those values don’t get “hit.”
This tells us that neither (6k +/– 1) value is a squared prime number, or a prime composite, which is the product of two or more prime numbers.
And thus, those two values that are not “hit” must both be prime numbers. Lastly, if both values for (k) are prime numbers, we know the two numbers must be a twin prime pair.
Returning to our example, if (p = 19), we find 2 pairs of twin primes: (419, 421); (431, 433).
Which satisfies the conjecture: p² < TwinP rime < (p² + 4p)
19² = 361 < (419, 421), (431, 433) < (361 + 4(19)) = 437
The table below compiles results for the first few primes of this modified Twin Prime dynamic.
Table 1.4
Column A : values for (x) = p — ((p +/ — 1) ÷ 3))
Column B : values for (y) = (p +/– 1) ÷ 3
Column C : the (+4p) range for prime (p), in terms (mod6) = (2p/3)
Column D : the number of primes < (p² + 4p)
Notice that the values in column A are always greater than the values of column B. Also note that the number of potential twin primes in column C is always ≥ twice the value of column D, which is the number of the lesser primes.
Now, let’s put all of this together to find a fresh pair of Twin Primes.
For our example, let’s consider the case for: (p = 17).
We begin with the Twin Prime Dynamic: p² < Twin Prime < (p² + 4p)
Which can be written as: 17² = 289 < Twin Prime < (17²+ 4(17)) = 357
(It seems notable to point out that another way to express (p² + 4p) is to calculate it as: (p + 2)² — 4.)
If (p = 17), the estimated number of possible locations for twin primes in the (+4p) range is calculated as: (2p/3) = 11.33333.
For an exact count of the number of values of the (+4p) range, in terms of (mod6), there are two equations:
If (p = 6k — 1), we calculate: (2/3p + 2/3) — 1
And if (p = 6k +1), then we calculate: (2/3p — 2/3)
For example, since 17 is of the form (6k — 1), we find that:
((2/3)(17) + 2/3) — 1 = 11
And thus, we find there are 11 values (mod6) between: 289 and 357.
Next, we determine how many times the “wavelengths” of the lesser primes “hit” the 11 values in that range.
To do that we calculate: √(p² + 4p)
If (p = 17), then we find that √(17² + 4(17) = √357 = 18.8944436
Which tells us the limit is: 17 itself; additionally, it tells us that there are 5 lesser primes of the form (6k +/– 1) such that: (5, 7, 11, 13, 17).
Next, we determine which of the 11 values (mod6) that those 5 lesser primes intersect with in the (+4p) range:
5 multiples of 5: (295, 305, 325, 335, 355)
3 multiples of 7: (301, 329, 343)
2 multiples of 11: (319 and 341)
2 multiples of 13: (299 and 325)
1 multiple of 17: (323)
Total: 13
In order to help visualize how the “wavelengths” of these lesser primes “hit” the different values for (k) along the number line (mod6), let’s check:
Next, we need to calculate the number of the “unique intersections.”
We find that (299, 301); (323, 325); and (341, 343), share values for (k), in terms of (mod6). This reduces our count. Such that, when we count up the values for (k) in the (+4p) range that are “hit” by the “wavelengths” of the multiples of the lesser primes, the new total is: 9 “uniquely hit” values.
This tells us that 9 of the 11 possible values (mod6) in the (+4p) range cannot be Twin Primes.
However, conversely, this also tells us that 2 of the 11 values (mod6) contain no prime products and no prime squares.
And thus, those two values for (k) must both be twin primes.
Thus for (p = 17), the related twin primes are: (311, 313) and (347, 349).
Which satisfies the dynamic of our modified twin prime conjecture:
17² = 289 < (311, 313) and (347, 349) < 17²+ 4(17) = 357
What follows below is a list of a few more examples, in terms (mod6).
Table 1.5
The columns, expressed in (mod6), from left to right: prime number (p); the (x, y) inflation rates for the prime number; the value for (p²); the values for Twin Primes in the (+4p) range; and lastly, the maximum value of that same range, (p²+ 4p).
To further understand why there is always a twin prime pair in the (+4p) range of this dynamic, let’s focus on the the number of (mod6) values in the range: (p²) to (p²+ 4p), which is determined by the prime (p).
For any prime (p) that can be expressed as: (6k — 1), the number of (mod6) values is determined by the equation: (p — (2k).
And for values of (p) which can be expressed as: (6k + 1), the number of (mod6) values is determined by the similar equation: ((p — (2k+ 1)).
Putting this all together into one chart, the Twin Prime dynamic becomes more self-evident:
Table 1.6
From left to right, the first columns tells us the number of primes less than the prime (p). Next, the second and third columns express in whole numbers, the values for (p²), and the values for: (p² + 4p). Next to that, expressed in (mod6), is the number of (6k +/– 1) values in that (+4p) range.
The three underlined columns are the number of (mod6) values in the (+4p) range, next to that is the number of (6k +/– 1) values that are “hit” by “waves” from the semi-arithmetic progressions of the primes less than (p). And finally, the last column, the numbers in red tells us how many twin primes there are in the (p² + 4p) range.
For a visual confirmation of the pattern of this semi-arithmetic progression, let’s create a (n x n) matrix for the primes and their multiples, expressed in (mod6) values. We track the semi-arithmetic progressions in terms of (x, y). For example:
5: (+3, +2, +3, +2, +3, +2…)
7: (+5, +2, +5, +2, +5, +2…)
The table below is a (n x n) matrix for primes of the form (6k +/– 1) less than 31. The diagonal in bold tracks the values, in terms (mod6), of the various primes squared, (p²):
Table 1.7
The red row is numbers of the form (6k +/– 1), and the blue column is the same. From left to right, and top to bottom, each row and column starts with a prime, or a prime composite of the form (6k +/– 1). The second column and second row counts the value (mod6) for that prime. The rest of the rows and columns track the arithmetic progression, in (mod6), of the (x, y) inflation rate of the prime number’s multiples, both horizontally and vertically.
Using this (n x n) matrix, since it corresponds to the values (6k +/– 1), in terms (mod6), any number that never appears in this chart is a twin prime.
Which means, that (n x n) matrix is a map of Twin Primes, by virtue of their absence.
Let’s consider a few examples of (n x n) matrices for primes and their multiples. For instance, let’s consider the case of (p = 11):
11² < Twin Primes < (11²+ 4(11))
Which is equivalent to: 121 < Twin Primes < 165
In terms (mod6), for (p = 11), the (+4p) range is between (20 and 27.5).
What values get “hit” by the lesser primes? The results are marked in pink.
Table 1.8
Clearly, evident from the (n x n) matrix above, there are plenty of (6k +/– 1) values where there are prime composites. But equally, there are also not enough lesser primes to “hit” all of the values (mod6) in that range.
Such that, we find the values (mod6) that are not “hit” are: 23, 25.
This can be expressed as: 20 < (23 and 25) < 27.5
Which can be rewritten: 121 < (137, 139) and (149, 151) < 165.
We find there is, indeed, at least one set of twin primes. In this case, there’s two.
What does it mean “there are not enough” lesser primes whose multiples can “hit” all the possible (mod6) locations? And how do we know the lesser primes will never hit all the (mod6) values? Why will certain values (mod6) for the multiples of the primes always be “skipped over”? To answer these questions, let’s return to the Sieve of Eratosthenes.
Using our modified number line and the Sieve of Eratosthenes, we can answer this in terms (mod6). The ”skipped over” values are due to the fact the multiples of the lesser primes alternate. And they do so according to a prime’s rate of its semi-arithmetic progression. Which, as we’ve shown, is dependent on a prime number’s unique values: (x, y).
As we’ve previously demonstrated, the alternating pattern for the multiples of the primes grows at an uneven rate: (+4p, +2p, +4p, +2p, +4p…). This oscillating rate of increase represents interactions of the closed set of numbers of the form (6k +/– 1). The set is closed by multiplication and by virtue of the Theorem of Arithmetic. In other words, the set of multiples of the lesser primes is limited by the fact that the addition and multiplication for the prime numbers and their multiples must be equivalent, and unique.
Additionally, the semi-arithmetic progressions of primes guarantees that certain values of the form (6k +/– 1) will always be “skipped over.” This is further confirmed when we consider the unique (x, y) values for a prime number. Plus, for any prime (p) the number of potential locations for twin primes in the (+4p) range is always greater than double the number of the lesser primes.
We also know that the multiples of the lesser primes that “hit” values in the (+4p) range reduce the number of potential spots that the lesser primes can “hit,” since some of the prime multiples “hit” the same values for (k). This leaves not enough of the lesser primes and their multiples to “hit” all the possible spots for twin prime pairs in the range: (p²) to (p² +4p).
For a proof by contradiction, let’s consider if the primes and their multiples marched not with their oscillating and uneven rate of their unique semi-arithmetic progression, but instead consider if they alternated regularly at an even and steady (+3p, +3p, +3p, +3p, +3p…) rate of progression.
Would that allow the lesser primes to “hit” all the (mod6) values?
Short answer: No. The reason is simple: because the multiples of the primes follow an even (+3p, +3p, +3p, +3p, +3p…) progression, that means the values for any prime of the form (6k +/– 1) would cycle through values divisible by 2. And thus, those multiples of the lesser primes would no longer be of the form (6k +/– 1).
Also if the common difference is (+3p), the set of multiples would no longer be a closed set under multiplication. Instead the resulting values would include multiples of +/– 1 (mod3).
For an example, let’s consider the case of (p = 7). If the pattern of the progression was: (+3p, +3p, +3p, +3p, +3p…), that leads to the alternating series of values:
Note how the steady (+3p) arithmetic progression “skips over” the prime composites that have the value (7)(6k — 1). Such as 35, 77, 119, etc.
That means this progression excludes the multiples of the form: (7)(6k + 1).
The oscillating values (+4p) and (+2p) represents the only way the lesser primes can and will “hit” all the multiples, and remain a closed set of prime composites of the form (6k +/– 1).
The reason is simple: for any prime (p), if you add (+4p) or (+2p), this follows the prime’s unique semi-arithmetic sequence, thus the result is always, and will only be, a multiple of the form: (6k +/– 1).
A Modular Proof Based on the Wave–like Nature of Primes and their Prime Products
Proof.
By focusing on the (mod6) nature of the prime numbers, their squares, and their subsequent prime composites, we can prove the infinitude of twin primes. We do this not by examining where primes are located on the number line but, rather, where the primes are not.
Definition: All primes > 3 are of the form: (6k +/– 1).
Definition: All prime squares are of the form: (6k + 1).
To prove our Twin Prime conjecture dynamic, let’s examine an example.
First, we must determine the (mod6) value for a prime number, we take:
((p +/– 1) ÷ 6)
For example, if (p = 11), then we know that: (11 + 1) ÷ 6 = 2
Next, we determine the unique (x, y) inflation rates for the semi-arithmetic progressions of a prime (p), in terms of (mod6).
x = ((p — (p +/– 1) ÷ 3))
y = ((p +/– 1) ÷ 3))
The difference, in terms (mod6), between the values for (11²) and (11) is (20–2 = 18). We can explain this with the semi-arithmetic progression (mod6) and the (x, y) values.
For (p = 11), the (x, y) rate is: (7, 4). And thus, it’s unique semi-arithmetic progression is: (2) + 7 + 4 + 7 = 20
This pattern of the (x, y) rates for the semi-arithmetic progressions holds for all primes (p). And it’s these uneven rates for the semi-arithmetic progressions that prove that the lesser primes can never and will never “hit” all the values (mod6) of the range: (p² + 4p).
To help visualize how this relates to the pattern of the semi- arithmetic progression of the prime numbers and the prime composites, let’s return to the Pascal-like Triangle. We can create an (n x n) matrix, in terms (mod6), and cut it in half and turn it on its side to create a pyramid.
The numbers in red are the values of the squares of (5, 7, 11, and 13). The diagonals are the (mod6) intersections for the (6k +/– 1) values. The resulting pyramid pattern depends on the (x, y) values.
As we can see below, if we add in one more term to the pyramid, the diagonals, as they descend from right to left, (or from left to right), now track the addition of the alternating values (x, y), such that:
Descending from right to left, the diagonals start with the value (k) for numbers of the form:(6k +/– 1), and the pattern alternates with the addition of the (x, y) values.
As evident in the pyramid above, certain values (mod6) are “skipped over.” Those same values, the ones that never appear in the pyramid no matter how far it is extended, they represent the values of (k) for pairs of twin primes.
In other words, if we were to extend the modified Pascal-like triangle to infinity, the pyramid would become a map of twin primes, by virtue of their absence.
Due to the nature of numbers of the form: (6k — 1) and (6k + 1) that share a value for (k), and the limit on the number of lesser primes which can combine to form prime composites, there can never be enough lesser primes to “hit” all of the possible values (mod6) for the potential twin primes in the (+4p) range.
The number of intersections of the “wavelengths” of the lesser primes is determined by and limited by the number of the lesser primes. As well, the number of intersections is also limited by a prime’s unique semi-arithmetic progression.
In terms of the prime (p), this means that for the lesser primes the common difference between terms in their semi-arithmetic sequences must hold to the pattern: (+4p, +2p, +4p, +2p…).
Each lesser prime’s semi-arithmetic progression is determined by the alternating values (+4p and +2p), which are relative to the prime’s “wavelength,” and its unique (x, y) values (mod6). Thus, it’s mathematically guaranteed that certain values will always be “skipped over.” And it’s equally certain that those “skipped over” values will always be twin prime pairs.
Altogether, this can be expressed:
For a prime (p), the number of values (mod6) of the range (+4p) = (2p/3). That value is always greater than twice the number of the primes < √(p² +4p). And that same dynamic holds true for any prime (p).
Additionally, the number of lesser primes for any prime (p) that intersect with its multiples in the range (+4p), will result in a number of “uniquely hit” values (mod6) in that same range (+4p).
When the number of values (mod6) that are “uniquely hit” is subtracted from the number of potential spots for twin primes in that same range, which we calculate as (2p/3), the remainder tells us the values (mod6) that are not prime composites or a squared prime number.
If they are numbers of the form (6k +/– 1), that are not squared prime numbers or the product of two (or more) lesser primes, then those numbers are proven to be prime. And if those same numbers are separated by a prime gap of (+2), they are twin primes.
In other words, as we climb ever higher up the number line, we will continue to encounter values of +/– 1 (mod6). When neither of those potential values are hit by the intersection of a “wavelength” of a lesser prime, we can confirm that the result must be a twin prime.
This dynamic is further confirmed by the Theorem of Arithmetic and the nature of the closed set under multiplication of the prime numbers and their prime multiples, all of which are numbers of the form (6k +/– 1).
The perpetual inequality between the lesser primes and their multiples, the “skipped over” values, and the number of possible twin primes, proves there will always be a pair of twin primes in the range:
p² < Twin Prime < (p² + 4p).
Consequently, this same dynamic proves that Polignac’s Twin Prime Conjecture is true.
There are indeed pairs of primes, with a gap of 2, continuing up to infinity.
Q.E.D.
