This work is licensed by its author, Gary William Croft, under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

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"In the ideal world of numbers and shapes, strange coincidences usually are clues that we're missing something. They Suggest the presence of hidden forces at work." – Steven Strogatz, *The Joy of X: A Guided Tour of Math from One to Infinity*

The first 1000 prime numbers are silently screaming: "Pay attention to us, for we hold the secret to the distribution of *all* primes!"

We heard the call, and with 'strange coincidences' leading the way have discovered compelling evidence that the 1000th prime number, **7919**, is the perfectly positioned cornerstone of a mathematical object with highly organized substructures and stunning reflectional symmetries. This object is dually enveloped by ** 89 ^{2} = 7921** and

Which leads us to point out that it's no accident this page's header image is of a 4-sided pyramid. What this iconic geometric form has to do with the first 1000 prime numbers will become evident toward the bottom of this page where we unearth the 'One Grand Pyramid' aka 'Tomb of the First 1000 Primes.'

For now we will be coy and state that the secret of the pyramid lies with two 4-times triangular numbers: The first, **2112** (4 x triangular number 528), equates to the total number of elements used to construct the pyramid, which is subdivided into its four lateral faces comprised of 528 elements each in the form of 32 cascading triangular numbers. And so it is that **2112**, index number of the 1000th prime as a member of this domain, is the terminating element of the pyramid bijecting 7919, i.e., 2112 and 7919 have a one-to-one correspondence.

The second pivotal number, **112**, is the element count of the pyramid's pyramidion or capstone. We interpret 112, which is 4 x triangular number 28 (the 2nd perfect number), as a 'string' reflecting the first three digits of the Fibonacci sequence, and representing the most fundamental of 'fundamental dyads,' 1+1=2. The index numbers of the set of natural numbers not divisible by 2, 3, or 5 populating each of the capstone's four faces incrementally cascade 7 triangular numbers (1, 3, 6, 10, 15, 21, 28) for a total element count of 28.

Introducing a 'capstone' into this model, as if alluding to the Great Pyramids of Giza and their sacred pyramidions (speculated to be gold covered), will be greeted with incredulity by many. That's fair. For the record, this model was developed without any conscious thought given to possible parallels with Egyptian pyramids until after the fact–and then only in the form of idle curiosity (which, by the way, we've yet to satisfy).

We want it known that the number 112 emerged relentlessly from the data. The fact that both 112 and 2112 are 4-times triangular numbers, and that 2112 is a palindromic form of 112 were clues prompting 100's of experiments that ultimately led to exposing 112's role in building a pyramidal geometry. Regardless, we believe that once you've 'grocked' the beautiful family of ratios, digital root symmetries and triangular number progressions your skepticism about equating a capstone with the distribution of prime numbers will retreat ...

We lay the foundation for the One Grand Pyramid by providing an exhaustive list of data points employing the Prime Spiral Sieve telescoped to modulo 90 and a corresponding 24 x 88 = 2112 cell matrix–the width reflecting the period-24 digital root of this domain, and 2112 being the index number of the 1000th prime number as a member of this sequence. Also note that every quadrant of the matrix (22 of 88 rows = 528 elements) is equivalent to one lateral face of our pyramid.

And here are a three more intriguing correspondences between 2112 (pyramid) and 112 (pyramidion): 1) The analytical matrix is 24 **x** 88 = 2112; while 24 **+** 88 = 112; 2) 2112/112/6 = 22/7 (a commonly used equivalent for pi) and it follows that 6(22/7)112 = 2112, while 528/28/6 = 132/7/6 = 22/7; and, 3) (11 + 13) x (41 + 47) = 2112, while 11+13+41+47 = 112.

It was the challenge of finding an object that will accommodate 2112 elements while preserving the digital root reflective symmetry fundamental to our defined domain (numbers not divisible by 2, 3, or 5) that led us to the perfect symmetry achieved when the 2112 elements are evenly distributed to the four equilateral sides of the One Grand Pyramid, configured in the form of counter-balancing polarities, with 2112/2 = 1056 pairs summing to 7920 (and note that 7920−1 = 7919, the 1000th prime, and 7920+1 = 89^{2}). And thus a rectangle is transformed into a beautifully proportioned pyramid.

Before getting to the data, the graphic below prefigures the pyramid to come:

And here are some tantalizing related factoids:

Plugging the 1000th prime, 7919, into the formula for triangular numbers, n(n+1)/2, gives us 7919(7919+1)/2 = 31,359,240 = 7919 x (109^{2} − 89^{2}), *and* every one of the One Grand Pyramid's thirty-two steps sums to a multiple of (109^{2} − 89^{2}) = 3960. (And, keep in mind that 3,6,9 and its permutations are fundamental to the rotating symmetry groups generated by this domain's factorization algorithms.)

The first three primes (2, 3, and 5) are set aside in this mathematical model (albeit accounted for in the grand total), given 1) they are the only prime numbers not members of our defined domain (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.

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

**7920** (which = 22 x 360 = 89^{2} − 1^{2})

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

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

**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 concatenating from another form of the 'fudamental dyad,' 11, aka Fibo_{2}, (viz., 11, 22, 33, 44, 55, 66, 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 number 24 and its square, **576**, play out dimensionally in this space are truly remarkable.

The diagram below captures the basic dimensions of our analytical matrix, followed by a rather long list of its intriguing features (which many of you will choose to skim). Then the fully populated matrix 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, or what might better be called Reflectional Symmetry Primes.

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

Also, be warned that you'll find a certain amount of annoying redundancy in the data compilation below. This was done deliberately to expose particularly interesting juxtapositions and associations. And note that **576** has been 'boldened' to emphasize its importance. And one can't help but wonder if there's a connection between the 88-unit dimension of this object and the 88 pitches of a modern piano, especially when you consider the connection the One Grand Pyramid and related Triangular Wave have to 3960 Hz–the 9th harmonic of fundamental frequency 440 Hz.

As you scan the list below you'll occasionally encounter what we're calling 'iterative digit division' (you can think of it as a corollary to 'iterative sum of digits' that defines digital root). For example, the cumulative sum of n not divisible by 2, 3, or 5 from 1 to the 1000th prime (7919) = 8,363,520. To calculate its iterative division root we get 8/3/6/3/5/20 = .00148148 ... (You'll note that, because one can't 'legally' divide by zero, all zeroes adhere to the digit preceding them.).

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. Thus, 1/89 = 1/10
^{2}+ 1/10^{3}+ 2/10^{4}+ 3/10^{5}+ 5/10^{6}+ 8/10^{7}+ 13/10^{8}+ ... . 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. - The 10th ranked perfect number (54 digits long, summing to 235) is generated by 89 in the formula: 2
^{(p-1)}* (2^{p}− 1), thus: 2^{(89-1)}* (2^{89}− 1) = 191561942608236107294793378084303638130997321548169216. - Total value of the 24 * 88 = 2112 cell matrix is 8,363,520, and average value is 3960 = 11 x 360 = 109
^{2}− 89^{2}. To show the depth of these relationships, we give you: 11 x (8/3/6/3/5/20) x 360^{2}= 2112 (7919's index number as a member of n not divisible by 2, 3, or 5). Related to this is (8/3/6/3/5/20) x 360^{2}= 192 = 24^{2}/3 = the total partitions of 10, and 4/8/4 x (8/3/6/3/5/20) x 360^{2}= 24 (index # of 89, the 24th prime, as a member of n not divisible by 2, 3, or 5, while 484 is the count of primes in the 242 'Mirror Prime Pairs," described and illustrated down this page. - F
_{11}(89): The decimal expansion of 89's reciprocal (1/89) is period-44 (see graphic below) composed of 22 bi-lateral 9 sums = 198, while 89 + 109 = 198, 7920/198 = 40 and 8,363,520/198 = 20 x 2112 (7919's index number as a member of this domain). And, curiously, 198's inverse (891) + 109 = 1000, while the sum of 89 and 109's inverses, 98 + 901, = 999. Then consider that, while it's obvious 997 of the first 1000 primes are not divisible by 2, 3, or 5, one might miss the fact that 997 minus its reverasl, 799, = 198 = 89 + 109. And for the record we note that 1/109's decimal expansion is period 108 (making it a 'long period prime' in that 1/p has the maximal period of p−1 digits). This period consists of 54 bilateral 9 sums = 486, which (coincidentally?) is the number of primes in the 243 pairs summing to 7920 (more about these, below).]

- As if balancing against 1/89's Fibonacci encodement, 89
^{2}= 7921 has its own hidden Fibonacci dimension:

- And the following study illustrates a period-24 palindromic cascade culminating in the Fibonacci digital root sequence (also period-24):

- 109
^{2}− 89^{2}= 3960 and 3960 x 2 = 7920; which equates to 8,363,520/(109^{2}− 89^{2}) = 2112, and when you plug 7919 into the formula for triangular numbers you generate 31,359,240 = 7919 x (109^{2}− 89^{2}). And here's another grouping that relates to these ratios: (67^{2}− 23^{2}) = (109^{2}− 89^{2}) and (67^{2}+ 109^{2}) − (23^{2}+ 89^{2}) = 7920 = 2(109^{2}− 89^{2}).

And here we correlate 11's additive sums with 3960, 7920 and the first 1000 prime numbers.

Diving deeper yet, here we show the absolute value delta differences of simple reversal sets in relation to primes 89 and 109 contextualized with the first 1000 prime numbers. Of course, we're well aware of the mantra, "Correlation is not causation." Regardless, these associations are profoundly suggestive of deep structural connections:

- (2 x 3
^{2}x 109) − (2 x 3^{2}x 89) = 360.

- Cumulative digital root of 7920 = 9600, while cumulative sum N = {0, 1, 2, 3, ...} (1 - 7919) = 31,359,240 = 7919 x (109
^{2}− 89^{2}) = 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 − (109
^{2}− 89^{2}) = 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. And, interestingly, (7919 x 7921) − (7919 + 7921) = 62,710,559, which is prime.

[Here we note our astonishment that π(62,710,560) = 3,713,160, and (again using iterative digital division) 3/7/1/3/1/60 x 112 (Fibo_{1-3}) x 7920 = 2112 (index number of the 1000th prime). How to explain that! ... especially considering the role 112 plays as the pyrimidion of our 2112 element 'One Grand Pyramid' housing the first 1000 primes, which you'll find at the bottom of this page.] - 90
^{2}− 89^{2}= 179 (prime); 90^{2}− 7919 = 181 (prime); twin primes 179 + 181 = 360. - 9+8+7+6+5+4+3+2+1+2+3+4+5+6+7+8+9 = 89.
- 10+9+8+7+6+5+4+3+2+1+2+3+4+5+6+7+8+9+10 = 109.
- 89 and its reversal require the most reversals and additions to become palindromic (steps required = 24, with the 'reverse-and-add' sequence cascading to the 13-digit palindrome: 8813200023188, with additive sum of 44, which happens to be the number of repeating digits in the decimal expoansion of 89's reciprocal).
- 89 is the 24th prime and 24th n not divisible by 2, 3, or 5.
- 24 is the 3rd 'Superprimorial,' viz. 1! x 2! x 3! = 24 and (1! x 2! x 3!)
^{2}= 1^{2}x 2^{2}x 3^{2}x 4^{2}=**576**. - F
_{4}(3) + F_{8}(21) = 24 (and both 3 and 21 are triangular numbers). - The prime numbers between 1 and 24 (2, 3, 5, 7, 11, 13, 17, 19, 23) sum to 100, while the first 24 counting numbers sum to 300 (triangular number T
_{24}). - 431, 433 are twin primes.(4 x 3 x 1) x (4 x 3 x 3) = 432.
- 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. - 8/30 x 5/24
^{2}x 432 = 1 ... thus 30/8 x 24^{2}/5/432 = 1. Also note that 24^{2}/4/3/2 = 24 and 24^{2}/7/9/2/1 x 112 = 512 = 2^{9}and 24^{2}/8/3/6/3/5/20 x 360^{2}= 4 x 432 = 3 x**576**) and 24^{2}/9/50/4 x 360^{2}= 96 x 432 = 72 x**576**).

- 24
^{2}=**576**. - (3/5/9)
^{2}x 360^{2}=**576**. - 1
^{2}x 2^{2}x 3^{2}x 4^{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 (a triangular number) = 2 x 3 x 5 x 7 (the 4th primorial).**576**+ its cumulative digital root (2880) = 3456 and 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 N = {0, 1, 2, 3, ...} from 1 to 2112 (Index # of the 1000th prime) = 2,231,328. (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 24
^{2}=**576**th prime member of this domain, 4211, has index # 1123 (Fibo_{1-4}and a prime), while the 24th triangular number (T_{24}) = 300 and 300 is the index number of 1123 (a prime). - Cumulative digital root sequence n 1, 2, 3, ... (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)**

(Curiously, and possibly not coincidences: 7920 (which = 22 x 360) x (22/360) = 484, and 7920^{2}/360^{2}= 484.)

Scroll down to find a "footprint" of all 242 of these pairs. - 8363520/242 = 34560 = 96 x 360 = 80 x 432 = 60 x
**576**= 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, i.e., 8/30 x 360 = 96.)
- 1/1/2 x (8/3/6/3/5/20) x 360 = 8/30; 1/1/2 x (8/3/6/3/5/20) x 360
^{2}= 96; 1/1/2 x (8/3/6/3/5/20) x 360^{3}= 1! x 2! x 3! x 4! x 5!. While, 8 x 3 x 6 x 3 x 5 x 20 x 576 = 1! x 2! x 3! x 4! x 5! x 6! (the 6th Superprimorial!)

- 8363520/360 = 23232 and twin primes, 4181759 + 4181761 = 8363520.
- 23232/360 * 30/8 = 242. (30/8 is ratio of numbers N = {0, 1, 2, 3, ...} to n not divisible by 2, 3, or 5).
- 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 sum N = {0, 1, 2, 3, ...} (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 x (24/88) =
**576**; 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) = 2n(n+1) (ref. oeis.org: A046092).
- 2111, 2113 are twin primes
- 2112's aliquot sum (the sum of its divisors) = 3984.
- 3 + 9 + 8 + 4 = 24; 3 x 9 x 8 x 4 = 432 x 2.
- 3/9/8/4 x 360 = 30/8 = 3.75 = 2
^{3}/(1^{2}+ 2^{2}+ 3^{2}+ 4^{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.
- (1
^{2}+ 2^{2}+ 3^{2}+ 4^{2}) x (1^{2}x 2^{2}x 3^{2}x 4^{2}) x 484 = 8363520. - 8,363,520/242 = 34,560.
- 3 x 4 x 5 x 6 = 360.
- 34,560 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. - 2
^{3}/(1^{2}+ 2^{2}+ 3^{2}+ 4^{2}) x 360^{2}= 1! x 2! x 3! x 4! x 5!. - 34,560/360 = 96.
- 2
^{3}/(1^{2}+ 2^{2}+ 3^{2}+ 4^{2}) x 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
- 2
^{2}x 3^{2}x 2^{2}x 3^{2}x 2^{2}= 5184 = 3^{2}x**576**. - 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 numbers N = {0, 1, 2, 3, ...} ) = 3960 (the average value of matrix).
- 1/1/2 x (8/3/6/3/5/20) x 360 = 8/30.
- 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.
- (8 x 3 x 6 x 3 x 5 x 20) / (9 x 50 x 4) = 24.
- 8363520/9504 = 880, while twin primes 197 + 199 x 24 = 9504, and 197 + 199 = 396.

- 7 x 9 x 1 x 9 = 567; 5 x 6 x 7 = 210 (a triangular number) = 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:

"Physicists have exalted symmetry to the position of *the* central concept in their attempts to organize and explain an otherwise bewildering and complex universe." Mario Livio, *The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry*.

Below is an image reflecting the "Mirror Prime Pairs" referenced above. Every pair is evenly divisible by 90 and has additive digital root sum = 9. Given the key role the number 11 plays in the structure of this matrix, it's likely not a coincidence that each side has 11^{2} + 11^{2} = 242 of these primes:

"Mathematics makes the invisible visible." – Keith Devlin, *The Language of Mathematics*

At the top of this page we declared that the first 1000 prime numbers are 'entombed' in a beautifully proportioned pyramid. To support this claim we offer the somewhat Archimedean construction below. We're tempted to call it the Tomb of the First 1000 Primes, but prefer the pun in 'One Grand Pyramid.'

As we've already alluded, to lay the foundation for a bijection with numbers not divisible by 2, 3, or 5, each of the pyramid's four lateral faces is constructed from a 32-step triangular number progression (oeis.org/A000217: a(n) = n(n+1)/2 ...). By definition, each triangular step is equilateral. In addition, every step dyad, starting with 1+3 = 4, sums to a perfect square, and thus each face of our pyramid cascades 1^{2} thru 32^{2}. (Note that 4 x 32 = 128 = the perimeter of the square base which has an area of 32^{2} = 1024 = 2^{10}. Which is to say that the pyramid's perimeter, 128, equals the number of triangular numbers employed in its construction.)

The four faces of our pyramid additively cascade 32 *four-times triangular numbers* (oeis.org/A046092: a(n) = 2(n+1) ...). These include Fibo_{1-3} equivalent 112 (rooted in T_{7} = 28; 28 x 4 = 112), which creates a pyramidion or capstone in our model, and 2112 (rooted in T_{32} = 528; 528 x 4 = 2112), which is the index number of the 1000th prime within our domain, and equals the total number of 'elements' used to construct the pyramid. Or, using the textbook way to visualize triangular numbers, 2112 = the total number of billiard balls filling the four faces, which in our case will be dually populated with natural numbers 1, 2, 3, ... and their associated numbers not divisible by 2, 3, or 5 in a 4-fold progression of perfect squares descending the faces of the pyramid.

The table below shows the telescopic progressions of triangular, 4-times triangular numbers and cascade of perfect squares that populate the pyramid's faces.

Both 112 and 2112 contain the 'fundamental dyad' 1+1 = 2 (which we featured earlier in this section). They give us a curious equation (employing iterative digital division), (2/1/1/2)(2112-112)(1/1/2) = 1000.

Speaking of iterative digital division–a powerful tool for exposing structure–we get this astonishing equation: iteratively dividing the digital roots of the first 12 Fibonacci numbers times the divisively iterated 1000th prime, 7919, times 360^{4} gives us 1000. Keep in mind that the first two and last two digits of the Fibo sequence below, 11 and 89, sum to 100; that 89 is the 11th Fibo number; that there are 1000 primes between 1 and 89^{2}; and that 89 has the Fibonacci sequence embedded in its decimal expansion:

**1/1/2/3/5/8/4/3/7/1/8/9 x 7/9/1/9 x 360 ^{4} = 1000**.

The graphic below shows the ranges of the numbers congruent to {1, 7, 11, 13, 17, 19, 23, 29} mod 30 populating each of the lateral faces along with their index numbers:

Next we focus on the pyramidion exclusively. We've populated it–and by extension this applies to the entire pyramid–in such a way as to achieve reflectional symmetry in the form of bilateral 7920 sums. The pyramidion is constructed from 56 such dyads, while the frustum accounts for the remaining 1000, for a total of 1056 x 7920 = 8,363,520 (which, of course, is the sum of the 2112 elements populating the pyramid).

The pyramid possesses 32 telescopically expanding steps (the first 7 of which constitute the pyramidion): The 1st (top) step has 4 elements, the 2nd 8 elements, the 3rd 12 elements, ... Given the symmetry of the bilateral dyads summing to 7920, and that 7920 = 15 x 528, it follows that every step of the pyramid sums to a multiple of triangular number 528. For example: Step 1 = 1 + 1981 + 7919 + 5939 = 15,840 = 7920 x 2 = 30 x 528, and amazingly 8,363,520 = 528 x the sum of Step 1, 15,840. And then we see that 56 x 7920 (additive sum of the pyramidion's 112 elements) = 443,520 = 528 x 840, while 15,840 x 28/528 = 840 and 7920 x 28/528 = 840/2. Also, 4/4/3/5/20 x 360^{2} = 432. And perhaps most interesting of all the pyramidion vs. pyramid ratios and proportions are these: 443,520 = 2 x 3 x 5 x 7 x 2112 and (8,363,520 − 443,520)/7920 = 1000, which equates to **((8,363,520 − (8,363,520 x 28/528))/((109 ^{2} − 89^{2}) x 2) = 1000** (and in relation to these, recall that

That said, this equation may be the most provocative of all:**1396 − (109 ^{2} − 89^{2})/10 = 1000.**

Why provocative? Because **1396** equals the total number of calculations required in our
Prime Number Counting Methodology to index all composite numbers in the range 1→89^{2}. Yet another connection to 89 is that the sum of the index numbers of every one of the 1056 dyads forming the pyramid sum to 2113, which is the index number of 7921 = 89^{2}.

To make the reflectional and rotational symmetry more conspicuous, we've converted the numbers not divisible by 2, 3, or 5 populating the pyramidion to their digital roots. When you rotate the opposing *horizontal* faces (as pictured) 90° counter-clockwise, and/or the *vertical* faces 90° clockwise, they superimpose with identical values. And we note that the additive sum of the pyramidion's digital roots = 504, which is consistent with the ratio 9504 x 28/528 = 504 (9504 being the cumulative digital root, 1 &rarrow; 7919).

Examine the above graphic and you'll discover four ways lateral and/or adjacent faces have bilateral 9 digital root sum symmetry (illustrated below):

Summarizing some of the above relationships:

8,363,520/443,520 = 2112/112 = 528/28 = 7/132.

28/528 x Pyramid Unit Sum = Pyramidion Unit Sum; 28/528 x Pyramid Face Unit Sum = Pyramidion Face Unit Sum (or Pyramidion/4).

528 x 28/528 = 28; 528 x 112/2112 = 28 (which happens to be the 2nd perfect number).

2112 x 28/528 = 112; 7/9/1/9 x 36^{2} = 112; and, 1/90 x 28 x 360 = 112.

24 x 88 = 2112; 24 + 88 = 112.

28/112 x 2112 = 528; and, 2112/112 x 28 = 528.

8,363,520/4 = 528 x (109^{2} − 89^{2}) = 528 x 3960.

3960 x 28/528 = 2 x 3 x 5 x 7.

500/28 x 112 = 2000 (frustum element count).

528/28 x 112 = 2112; 2112/112 x (7/9/1/9) x 36^{2} = 2112; 112/(7/132) = 2112; and, 1/90 x 528 x 360 = 2112.

112((109^{2} − 89^{2})/(2 x 3 x 5 x 7)) = 2112.

(109^{2} − 89^{2})/(528/28)/(2 x 3 x 5 x 7) = 1.

6π x 112 = 2111.1526321, while 6(3 1/7) x 112 = 2112 (3 1/7 is a commonly used approximation of pi and 528/28/6 = 3 1/7 and 528/6 = 2112/24).

28/528 x 2112/112 = 1.

And, we note that:

(2112-112)/2 = 1000.

(2112-112) x 1/1/2 = 1000.

500/28 x 112/2 = 1000.

2000/112 x (7/9/1/9) x 36^{2}/2 = 1000

(2/1/1/2)(2112-112)(1/1/2) = 1000.

(2112 − (2112 x 28/528)/2) = 1000.

(8,363,520 − (8,363,520 x 28/528))/7920 = 1000.

(7 x (4/3/2)^{2})^{2} x (7 x (5/7/6)^{2}) x (3/60)^{2} x 360^{3}/112 = 1000 (whew!).

1/2/3/4/5/6/7/8 x 112 x 360 = 1.

7/11 x 2112/112 = 12.

1/1/2 x (8/3/6/3/5/20) x 360 = 8/30 = ratio of numbers not divisible by 2, 3, or 5 to N = {0,1,2,3, ...}.

1/1/2 x (8/3/6/3/5/20) x 360^{2} = 96 = the number of perfect squares generated by the Prime Spiral Sieve during one complete modulo 90 factorization cycle.

112 x (109^{2} − 89^{2}) = 443,520 (pyramidion's cumulative sum n not divisible by 2, 3, or 5).

2112 x (109^{2} − 89^{2}) = 8,363,520 (pyramid's cumulative sum n not divisible by 2, 3, or 5).

1/1/2 x (8/3/6/3/5/20) x 360^{3} = the 5th 'Superfactorial = 1! x 2! x 3! x 4! x 5! = 34560.

2/1/1/2 x (8/3/6/3/5/20) x 360^{3} = 2 x 34560.

Scanning the above data, the eagle-eyed will have noticed this distinctive pattern. It speaks for itself:

528 ... 28, as in 528 x 28/528 = 28;

1056 ... 56, as in 1056 x 28/528 = 56;

2112 ... 112, as in 2112 x 28/528 = 112;

9504 ... 504, as in 9504 x 28/528 = 504;

23232 ... 232, as in 23232 x 28/528 = 1,232;

69696 ... 696, as in 69696 x 28/528 = 3,696;

8,363,520 ... 3,520, as in 8,363,520 x 28/528 = 443,520.

"The triangular numbers have important implications within both our everyday world and the world of number theory." – Edward B. Burger, PhD, *An Introduction to Number Theory*

Speculating beyond the pyramidal model just described, the ratios seem to suggest that this geometry can be conceived sinusoidally as a Fourier series forming continuous triangular waves that reverse polarity in quarter cycles.

For example, the 9th harmonic of the fundamental frequency 440 Hz = 3960 Hz (and keep in mind that 3960 = 109^{2} − 89^{2}, their relationship to the first 1000 primes covered in detail earlier in this section). Then consider that 8,363,520 (additive sum of the pyramid)/(109^{2} − 89^{2}) = 2112 (index # of the 1000th prime), and that 443,520 (additive sum of the pyramidion)/(109^{2} − 89^{2}) = 112 (index # of 419, the 81st prime [as in 9^{2}, interestingly], and in turn 7919 x 28/528 = [419]; whole number part taken).

Here's a draft of what the proposed triangular wave might look like:

The relationship between the triangular numbers and Fibo_{1-3}, 112 (the element count of our pyramidion), is further illuminated by this period-30, modulo 8 generated palindrome:

And the following 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, where 243 x 7920 gives additive sum **1,924,560** (and we note that 7920 = 22 x 360 = 2(109^{2} − 89^{2}), which equates to 1,924,560/(2(109^{2} − 89^{2}) = 243.).

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.

Here we incorporate the formula for k = the cumulative sum (1-n) for n ≡ {29} modulo 30; namely, **k = (n+1) ^{2} * 8/30/2**.

**[(7919+1) ^{2} * 8/30/2]/2112/11 = 360.**

In the following provocative study 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 find the following fascinating relationships:

Here are some thought provoking coincidences:

Then there are these strange coincidences:

And here's a related formulation that begs further study: [Note: In the equation illustrated below, where (89 * 8/30 +1) * (89 +1)/2 = 1113, the whole number part is not taken for (89 * 8/30 +1), unlike in the formula shown for index numbers of n not divisible by 2, 3, or 5, where whole number parts *are indeed* taken, e.g., 89's index number equates to 89 * 8/30 +1 = 24.]

Here we connect the power series we presented on our home page involving the first three primes to the first 1000 primes in toto:

And here, at the risk of making fools of ourselves, we leave you with an equation full of sound and fury, signifying nothing:

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