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The first 1000 prime numbers are quietly screaming: "Pay attention to us, because we hold the secret to the distribution of *all* prime numbers." We will provide what we believe is compelling evidence that the 1000th prime number is the perfectly positioned cornerstone of a rectangular shaped geometry with highly organized substructures dually enveloped by 89^{2} and 7920.

Let's proceed with our case:

For starters, 2, 3, and 5, the first three primes, although obviously counted among the first 1000, are set aside for the purpose of this analysis (albeit accounted for in the total), given 1) they are the only prime numbers not members of our defined factorization domain, i.e., not members of the set of natural numbers ≡ {1, 7, 11, 13, 17, 19, 23, 29} modulo 30, and b) they and their primorial, 30, are nonetheless present given they form the structure within which the 8-dimension (mod 30) and 24-dimension (mod 90) factorization algorithms described on this site operate.

We will use a matrix as an analytical tool to reveal structural symmetries specific to the first 1000 prime numbers. The matrix in question has 24 x 88 = 2112 cells, and thus 2112 is 7919's index number as a member of this domain (and note the Fibonacci palindrome). It is populated, as we've suggested, by natural numbers not divisible by 2, 3, or 5.

Our specified range for this analysis begins with 1 and terminates with 7919 (the 1000th prime number), and is circumscribed by two numbers:

**7920** (which = 22 x 360)

and **7921 **(which = 89^{2})

(using the standard notation to represent counts of primes from 1 to n):

**π(7920) = 1000** and **π(89 ^{2}) = 1000.***

*The reason we’ve presented two (seemingly redundant) counting ranges will become evident.

Study the relationships detailed below and you'll discover a palindromic spine (viz., 11, 22, 33, 44, 55, 88, 121, 242, 343, 484, 2112, 23232, 69696), vertical and horizontal reflectional symmetry composed of prime numbers (especially, the 242 'Mirror Prime Pairs'), and other features that suggest this 'mathematical object' is worthy of further study. For example: The myriad ways that numbers 24 and its square, **576**, play out dimensionally in this space are truly remarkable.

The diagram below captures the basic dimensions of this object, followed by a rather long list of its fascinating features. Then the matrix itself is presented, showing row and additive sum totals for primes (hi-lited in black). After that, you'll find a graphical "footprint" of the 484 prime numbers between 1 and 7920 (exactly 11^{2} + 11^{2} on each side!) that form 242 Mirror Prime Pairs.

[Note:The cumulative sum and indexing formulae referenced below can be found on a page dedicated to them, here: **
Cumulative Sum and Indexing Formulae**.]

Here's a quote to keep in mind while scanning these data:

"Mathematicians have long wondered at the haphazard way in which the primes are sprinkled along the number line from one to infinity, and universality offers a clue. Some think there may be a matrix underlying the Riemann zeta function that is complex and correlated enough to exhibit universality. Discovering such a matrix would have 'big implications' for finally understanding the distribution of the primes ..." Natalie Wolchover, "In Mysterious Pattern, Math and Nature Converge,"
*Quanta Magazine,* 02.05.2013

- F
_{10}(55) * F_{12}(144) = 7920. - F
_{10}(55) * F_{12}(144) − F_{1}(1) = 7919 (the 1000th prime number). - F
_{10}(55) * F_{12}(144) + F_{1}(1) = 7921 = 89^{2}(and every 11th Fibonacci number, starting with F_{11}, has 89 as one of its divisors). - The additive sums of Fibonacci numbers sequentially divided by 10 raised to the power of their sequence numbers equate to the reciprocal of 89. Similarly, the additive sums of Fibonacci numbers divided by 10 raised to the power of 109
*minus*their index numbers produce the reciprocal of 109. - Total value of the 24 * 88 = 2112 cell matrix is 8363520, and average value is 3960, while 3960/360 = 11.
- F
_{11}(89): The decimal expansion of 89's reciprocal (1/89) is period-44, composed of 22 bi-lateral 9 sums = 198, while 7920/198 = 40 and 8363520/198 = 20 x 2112 (7919's index number as a member of this domain). - 109
^{2}− 89^{2}= 3960 and 3960 x 2 = 7920; which equates to 8363520/(109^{2}− 89^{2}) = 2112, and 89^{4}− 7919^{2}= 8 x 3960. - (2 x 3
^{2}x 109) − (2 x 3^{2}x 89) = 360. - Cumulative digital root of 7920 = 39600, while cumulative sum n {1, 2, 3, ...} (1 - 7919) = 31359240 = 7919 x 3960 = 11 x 360 x 7919.
- 8 x 3 x 6 x 3 x 5 x 2 = 4320, and 4 x 3 x 2 = 24.
- 4320 − 3960 = 360.
- 89's index # as a member of this domain = 24, while 89
^{2}= 7921 has index #2113 (a twin prime, with 2111). - 89/90
^{2}= .010987654321. - The first 89 digits of pi's decimal expansion sum to 432.
- (89 x 91) − (89 + 91) = 7919.
- 90
^{2}− 89^{2}= 179 (prime); 90^{2}− 7919 = 181 (prime); twin primes 179 + 181 = 360. - 89 and its reversal require the most reversals and additions to become palindromic (steps required = 24).
- 9+8+7+6+5+4+3+2+1+2+3+4+5+6+7+8+9 = 89.
- 89 is the 24th prime and 24th n not divisible by 2, 3, or 5.
- F
_{4}(3) + F_{8}(21) = 24. - The prime numbers between 1 and 24 (2, 3, 5, 7, 11, 13, 17, 19, 23) sum to 100, while the first 24 integers sum to 300.
- 431, 433 are twin primes.(4 x 3 x 1) x (4 x 3 x 3) = 432 (parens for emphasis).
- 7919 ≡ 431 (mod
**576**), while 7921 (or 89^{2}) ≡ 433 (mod**576**), and, of course, 7920 ≡ 432 (mod**576**). - F
_{10}(55) + F_{14}(377) = 432. - 24
^{2}=**576**. - F
_{8}(21) + F_{9}(34) + F_{10}(55) + F_{11}(89) + F_{12}(144) + F_{13}(233) =**576**. **576**= 100_{10}_{24}**576**= 484_{10}_{11}**576**= number of modulo 90 factorization dyads and associated {9/3} star polygons that algorithmically account for all composite numbers > 5 not divisible by 2, 3, or 5.**5 x 7 x 6**= 210 = 2 x 3 x 5 x 7 (the 4th primorial).**576**+ its cumulative digital root (2880) = 3456.- 3 x 4 x 5 x 6 = 360.
- F
_{24}= 46368. - 4 x 6 x 3 x 6 x 8 = 3456.
- The cumulative sum of natural numbers {1, 2, 3, ..} from 1 to 2112 (Index # of the 1000th prime) = 2231328. (Here is the formula for the cumulative sum (k) of n 1, 2, 3, ...:
**k = (n + n**.)^{2})/2 - 2 x 2 x 3 x 1 x 3 x 2 x 8 =
**576**. **576**= number of 4 x 4 Latin Squares.- In 2016, Ukrainian mathematician Maryna Viazovska published two papers (the 2nd in collaboration with associates) proving that the densest packing of congruent spheres in Euclidean space is in dimensions
**8**and**24**; here's the theorem statement for the latter: "Theorem 1.1. The Leech lattice achieves the optimal sphere packing density in R^{24}, and it is the only periodic packing in R^{24}with that density, up to scaling and isometries." Here's a link to the 2nd paper: "The sphere packing problem in dimension 24". - The
**576**th prime member of this domain, 4211, has index # 1123 (1st 4 Fibonacci numbers). - Cumulative digital root sequence (1 - 7920) contains 345 prime numbers; 345's divisors (1, 3, 5, 15, 23, 69, 115, 345) =
**576**. **576**_{10}= 484_{11}.- 484 = number of primes in range (1 - 7920) that form 242 'Mirror Prime Pairs.'*

* prime pairs in each period-24 segment of our domain with summands that spatially 'mirror' opposite the center-line of a 24-wide matrix. All such pairs are evenly divisible by 90. For example, the first row contains 9 such pairs (working from the center outward):

**(43 + 47); (37 + 53); (31 + 59); (29 + 61); (23 + 67); (19 + 71); (17 + 73); (11 + 79); and (7 + 83)**(Scroll down to find a "footprint" of all 242 of these pairs.)

- 8363520/242 = 34560 = 96 x 360 = 80 x 432 = 60 x 576 (and note that 34560 = 1! x 2! x 3! x 4! x 5!, the 5th 'superfactorial').
- 8363520/96 = 242 x 360. (96 is the 360° term periodicity for n not divisible by 2, 3, or 5.)
- 8363520/360 = 23232.
- 23232/360 * 30/8 = 242. (30/8 is ratio of natural numbers 1, 2, 3, ... to n not divisible by 2, 3, or 5).
- A day on Venus is equivalent to 242 Earth days. 365 - 242 = 123.
- Entire matrix consists of 1056 dyads summing to 7920. Of these, 243 form prime pairs. 243 = 9
^{2}+ 9^{2}+ 9^{2}= 3^{5}. - Cumulative natural number sum (1 - 7920) = 31367160 = 11 x 89
^{2}x 360 = 2^{2}x 5 x 89^{2}x 198. - 3 x 1 x 3 x 6 x 7 x 1 x 6 = 4 x 567 while 7 x 9 x 1 x 9 = 567.
- Our mathematical object, a matrix, is 24 x 88 = 2112 cells.
- 2112/(88/24) =
**576**. - 2112/8
^{2}= 33. - (2112/8
^{2}) − (**576**/8^{2}) = 24. - 2112 is the sum of 2 distinct powers of 2, i.e., 2
^{6}+ 2^{11}= 2112 (ref. oeis.org: A018900). - 2112 is a 4 times triangular number, defined as a(n) = 2 x n x n x (n+1),

or 2 x 2112 = 4224; 4224 x 2112 = 8921088 =**576**x 2^{7}x 11^{2}; 8921088 x (2112 + 1) = 18850258944 =**576**x 2^{7}x 11^{2}x 2113 (ref. oeis.org: A046092). - 2111, 2113 are twin primes
- 2112's aliquot sum (the sum of its divisors) = 3984.
- 3 x 9 x 8 x 4 = 432 x 2.
- 2112/96 = 242 (242 = number of Mirror Prime Pairs and 96 is the 360 degree digital root Fibonacci periodicity when indexed to n not divisible by 2, 3, or 5, i.e., period 32 every 120 degrees).
- 2((8363520/(8x9x10x11)) = 2112.
- 8363520/242 = 34560.
- 3 x 4 x 5 x 6 = 360.
- 34560 is the 5th "superfactorial," i.e., the product of first 5 factorials, viz. 1! x 2! x 3! x 4! x 5! = 1 x 2 x 6 x 24 x 120 = 34560 (ref: oeis.org: A000178). Curiously, the 6th superfactorial (24883200) =
**576*** 432 * 100. - 34560/360 = 96.
- 34560/432 = 80.
- 34560/
**576**= 60. - 8 x 3 x 6 x 3 x 5 x 2 = 4320 = 10 x 432 = 12 x 360.
- 8363520/360 = 23232
- 23232/2112 = 11.
- 23232/(8/30) = 242 * 360.
- 2112/192 = 11.
- (191,193) are twin primes; 191 + 192 + 193 =
**576**. - 192 = total number of partitions of 10.
- 192 = 5 + 7 + 11 + 13 + 17 + 19 +23 + 29 + 31 + 37 (10 consecutive primes).
- 8353520 x 30/8 = 31363200 = 242 x 360
^{2}. - 8363520 x 30/8 + 3960 = cumulative sum (1 - 7920) = 31367160, while 31367160 = 360 x 11 x 89
^{2}. - 1056 (number of dyads summing to 7920) x 8/30 (ratio of numbers not divisible by 2, 3, or 5 to natural numbers, 1, 2, 3, ...) = 3960 (the average value of matrix).
- Cumulative digital root compiled for natural numbers congruent to {1, 7, 11, 13, 17, 19, 23, 29} mod 30 from 1 to the 1000th prime, 7919 = 9504.
- 9504 = 22 x 432 = 24 x 396 = 88 x 108 = 2
^{5}x 3^{3}x 11. - 9504/1056 (number of pairs in our domain summing to 7920) = 9.
- 8363520/9504 = 880.

- 7 x 9 x 1 x 9 = 567; 5 x 6 x 7 = 210 = 2 x 3 x 5 x 7.
- Of the 1000 primes between 1 and 7920, 997 are n not divisible by 2, 3, or 5 ... 9 x 9 x 7 = 567.
- In physics, 11 is the number of dimensions in M-theory "that unifies all consistent versions of superstring theory."
- 11 is not only the Fundamental Repunit. It's also the Fundamental Dyad (1 + 1 = 2) and the Fundamental Palindrome! And, of course, it's also the first palindromic prime and 1 + 1 = 2 ... initiates the Fibonacci sequence. And should we be surprised that the 'Magic Angle' for superconductivity (a precipitous drop in electrical resistance) when two sheets of graphene are stacked and twisted at a relative angle, is 1.1°! Lastly, when we add 11 + 89, we score 100!

Here is the matrix under discussion:

Below is an image reflecting the "Mirror Prime Pairs" referenced above. Astounding that each side of the matrix has 11^{2} + 11^{2} = 242 of these primes:

And here are the same primes identified by number:

Here's a tantalizing example of how Cumulative aka Additive Sum Formulae can be employed as analytical tools to reveal structure; in this case, exposing a beautiful bidirectional progression at the heart of the mathematical object we've been describing above:

And this example also exposes hidden structure:

**π(7920) = 1000**; 7919 is 1000th Prime.

There are 243 prime pairs among the first 1000 primes that sum to 7920, with additive sum 1,924,560.

Additive Sum n 1, 2, 3, ... (1 - 7919) = 31,359,240.

Additive Sum n ≡ (1, 7, 11, 13, 17, 19, 23, 29} modulo 30 (1 - 7919) = 8,363,520.

Now consider this:

1 x 9 x 2 x 4 x 5 x 60 = 21,600 = 60 x 360 = 50 x 432.

3 x 1 x 3 x 5 x 9 x 2 x 40 = 32,400 = 90 x 360 = 75 x 432.

8 x 3 x 6 x 3 x 5 x 20 = 43,200 = 120 x 360 = 100 x 432.

We leave this section with a provocative study, where we sieve the delta differences in prime number counts between the squares of n congruent to {89} modulo 90 and test their divisibility by 360 and 1000 (and we find especially interesting those deltas evenly divisible by both):

Focusing in on the deltas with divisibility by 1000, we shown the following fascinating relationships:

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