BS Rule 161: The even numbers of the 1^st Diagonal leg of the Brooks (Base) Square fall into two groups, excluded and included:

• the included even #s ... 4-8-12-16-20-... are both on the upper 1^st Diagonal, present in the Strict Inner Grid (SIG) and all are divisible by 4;
• the excluded even #s ... 2-6-10-14-18-... are only on the upper 1^st Diagonal, excluded from the SIG and not divisible by 4; 
• the excluded #s, when divided by 2 equal the odd numbers ... 1-3-5-7-... of the other 1^st Diagonal, of odd number and below the PD. When multiplied by 2, they become a subset of the included evens above;
• both excluded and included #s have 1^st # values of 2-6-0-4-8;
• both excluded and included #s  have 2^nd # values that fall into a pattern of clusters containing a 2:3 ratio of the lower to higher 2^nd # values, e.g. 22-26 | 30-34-38, and, 12-16 | 20-24-28 ;
• the sums, ∑, of the excluded even #s grow as: 50-150-250-...;
• the sums, ∑, of the included even #s grow as: 100-200-300-...;
• both excluded and included 2^nd # values increase by 10 when going across the chart.

Note:

BS Rule 162: The interconnection of the ISL and the Fibonacci number series, Ϝ, can be seen at the start, with the number one. The ISL, built on the squares of whole number integers ... 0-1-2-3-4-5-... begins with 1. The Golden Rectangle, ⎄, whose long side is the Golden Mean,  Φ=1.618, has a short side of 1. The embedded 1x1 square has a diagonal (hypotenuse) of the square root of two, 2^½. The hypotenuse value of a square with sides of ΦxΦ will equal 2^½Φ, that of a square Φ²xΦ² will equal 2^½Φ², that of Φ³xΦ³ will equal 2^½Φ³, and so on. Again, the difference, ∆, between successive square’s hypotenuses increasing by the product of Φ and the previous Φⁿ value is simply an increase in the exponential value of Φ as Φ, Φ², Φ³, Φ⁴,...

Note: Example: 144xΦ=233; 233xΦ=377; 377xΦ=610;

 144xΦ²=377, 144xΦ³=610

BS Rule 163: While both the ISL and the Fibonacci number series, Ϝ, show the difference, ∆, of the successive squares as having a 3-5-7-9-... progression. This is a superficial similarity in that the ISL series describes a specific order in the progression and is valid only for that segment, while in the Ϝ series, the difference, ∆, between the successive squares is actually a constant throughout the progression, i.e. at any point in the series, the ∆ between the successive squares at that point is the same constant ... Φ² ... this being based on the same fundamental core ∆ between each side (#) of the square ... that of Φ.

Note: The ISL increases sequentially as the odd-number summation series: (1-2-3-4-5-...)²=1-4-9-16-25-... and differences, ∆, of (1)-3-5-7-9-... respectively. The Ϝ series increases, not additively as in the ISL, but by a constant factor of Φ squared, or Φ², as the sides are increased exponentially as: Φ-Φ²-Φ³-... (and not as Φ-2Φ-3Φ).

BS Rule 164: The Ϝ series progresses as the exponential squares of the fixed Φ unit. Nevertheless, the difference, ∆, between successive square steps follows as Φ³-Φ⁵-Φ⁷-Φ⁹-... an odd number series of exponential Φ values. In this respect, the Ϝ squares increase in exponential values of Φ as Φ-Φ²-Φ³-Φ⁴-Φ⁵-..., their respective areas increase as Φ²-Φ⁴-Φ⁶-Φ⁸-Φ¹⁰-..., respectively, and, the ∆ between the successive areas follows as Φ³-Φ⁵-Φ⁷-Φ⁹, ..., respectively.

Note: The difference between successive Ϝ #s is Φ...  between every other Ϝ # it is Φ²... between every third Ϝ # it is Φ³. In the charts below, we see that the differences between Φ-Φ²-Φ³-... is itself yet another Ϝ-like series. The difference, ∆, between the subsequent Φⁿ values follows exactly as a Ϝ series. Each subsequent value of Φⁿ is the sum,∑, of the two preceding values.

Another way to look at this is the progression of squares, as above, where the sides grow as Φ-Φ²-Φ³-.... In actuality, as:

Φ¹=0+1Φ

Φ²=1+1Φ

Φ³=1+2Φ

Φ⁴=2+3Φ

Φ⁵=3+5Φ

Φ⁶=5+8Φ

Φ⁷=8+13Φ

Φ⁸=13+21Φ

Every new side is the sum, ∑, of the two preceding sides.

BS Rule 165: There is a natural repetitive cycling of one even for every two odd # values ... as even-odd-odd-even-odd-odd,... within the Ϝ series. This corresponds to spiraling through the two odd and one even 1^st Diagonals on the Brooks (Base) Square that has been modified to include the even 1^st Diagonal + both odd 1^st Diagonals (as in Base Square).

Every even #, excluded and included, alternately, that is included in the Ϝ series is Φ³ (approximately 2^½x3) larger or smaller than its next or previous even #, respectively.

Note:

BS Rule 166: The difference, ∆, between the spiraling square steps of the ISL and that of the Ϝ series is distinct. The ISL is a relatively slow, tight winding of a straightforward linear progression of areas, adding the sums of the previous square area to the next area, to form the new square. The Ϝ series is a fast winding of an exponential geometric progression of areas formed by a similar process of the addition of areas, only the areas are themselves exponential squares ... squares of the squares, if you will ... and not simple geometric squares as in the ISL. The growth of the Ϝ series squares is vastly superior to that of the ISL, reaching six figures in 24 steps as 24/3=8 turns of the spiral.

Note: 

• A square with sides of 1 has a hypotenuse of 2^½.
• A square with sides of 3, has a hypotenuse of 4.2426=2^½x3.

1.6180339887 x 1.6180339887 x 1.6180339887 = 4.23606798 Φ³=4.2358. Close, but not exact. 

• A square with sides of Φ has a hypotenuse of 2^½xΦ=2.288=[(Φ)²+(Φ)²]^½.

BS Rule 167: Take any Fibonacci, Ϝ, # and divide it sequentially by the Golden Mean, Φ, and it will resolve down to 1.17105. (1.17105)²=1.37137.

The Fine Structure Constant (FSC), or α, equals approximately 1/137. The FSC is one of the pure number, dimensionless universal constants of physics. It defines the fundamental interaction ... coupling constant (strength)... of the electromagnetic force (EMF). (See “GoDNA: The Geometry of DNA”  to see how 1.37(α) and 1.618 (Φ) are fundamental in the DNA Master Table ... defining the geometric structure of the DNA double helix in the axial view.)

The FSC is actually about 1/137. It simply represents a pure dimensionless # that functions as a proportion ... a ratio factor ... in the mathematical description of other dimension-based physical parameters that yield the coupling strength of the EM interaction. It is extremely intriguing that 1.37 shows up here in the Ϝ series, but for now, we will call it a speculative interconnection.

Note: Example: 144/Φ⁴=1.17105   (1.17105)²=1.37137 

BS Rule 168: Take any Fibonacci, Ϝ, # and divide it sequentially by the Golden Mean, Φ, and it will resolve one step further down, past 1.17105, and you will get numbers scattered around 0.723, the inverse of which is close to 1.38. The exception is number 5, it resolves to 0.7295515, with an inverse # of 1.3707052. Coincidence?

BS Rule 169: The solution to the strong form of the Goldbach Conjecture ... as modified by Euler ... “that every even positive integer greater than or equal to 4 can be written as a sum of two primes” is presented here as the Periodic Table of Primes (PTOP), an unequivocal pattern of primes, in sets of two, which account for every even positive whole number integer. (Please donate any prize money associated with this proof to the help, nutrition, education, shelter and prevention of abuse of the underprivileged children of the world.)

A simple graphical plot of the pairs of prime numbers whose sum, ∑, equals the next sequential even whole number integer ... 4-6-8-10-12-14-16-... reveals a series of patterns amongst those prime pairs:

• horizontally, across the grid left to right, one or more prime pair sets will form the even number and the first prime number will increase while the second prime number decreases;

• vertically, down each column of prime pairs, the second prime number of that pair set, after 2, increases exactly in the order of the primes from 3 to infinity;

• diagonally, from top left to bottom right, the first prime number of that pair set, after 2,  increases exactly in the order of the primes from 3 to infinity;

• diagonally, from top left to bottom right, the actual number of prime pairs that can form a given even number, after 2, starts with 1 and increases  by 1, in strict numerical sequence as: 1-2-3-4-5-6... the progressive sum of these prime pairs being designated as ∑#PP;

• combining the information about the prime pairs (PP) pattern distribution above, one can use the PTOP to help determine the “next” prime, from the smallest to the largest, by simply extending the PP pattern into the next corresponding even number ... or the one thereafter ... or the one thereafter, and so on ... until the pattern is fulfilled. Double-checking that the horizontal, vertical and diagonal pattern requirements are all met, you will now have assurance that that “next” prime number is indeed, prime. In doing so, you will also reconfirm the validity of the solution of Euler’s reformulated version ... a.k.a. the “strong” version ... of the Goldbach Conjecture. The conjecture has been confirmed up to 4x10¹⁴ numbers.

Note: The sum, ∑, of the actual numbers of prime pair (PP) sets in the first column ... ∑#PP ... increases by one, giving the sequence: 1-2-3-.... These PP sets run diagonally from top-left to bottom-right. We are simply adding one new PP set. Note that for any given set, the first prime # of that set increases in strict prime sequence order (no primes are missing!), while the second # remains fixed, e.g. 3+7, 5+7, 7+7. The sums, ∑, of the PP sets grow as one goes down the diagonal. The empty blank spaces going down diagonally, while not being prime, do follow the strict odd number sequence ... as if they were there ... acting as place holders, e.g. 3+11, 5+11, 7+11 and 11+11 has a blank diagonal space where 9+11 would go if 9 were prime, but it’s not, nevertheless, it acts as a blank placeholder for the odd number sequence, and the next valid prime number 11 is over and down diagonally 2 spaces from its predecessor, 7. Restated: If all the odd numbers were prime then the diagonals would all line up, corner to corner, to give a straight line; but for those odd numbers which are not prime, the diagonal corners are staggered with blanks stacked for each non-prime odd number. See the PTOP.

Going down vertically from the top, this time it is the second prime # of that set increases in strict prime sequence order (no primes are missing!), while the first # remains fixed, e.g. 5+5, 5+7, 5+11,.... The sums, ∑, of the PP sets grow as one goes down the column. The empty blank spaces going down vertically, while not being prime, do follow the strict odd number sequence ... as if they were there ... acting as place holders, e.g. above the 5+9 set is blank as 9 is not prime, but it does follow the odd number sequence as 5-7-9-11.

Going horizontally across, the prime #s of the PP sets both change. The first prime # increases, while the second decreases. And while they both do respect the natural sequence of primes, they do NOT do so in strict order, i.e. there are jumps along the sequence, bypassing some of those primes that are in strict order sequence, e.g. 3+19, 5+17, 11+11 (forming ∑=22), the first prime #s of the PP sets are 3-5-11 ... missing the 7 between the 5 and 11, and the second prime #s going in the reverse direction are 11-17-19 ... missing the 13 between the 11 and 17, and if you look at the chart there is a blank space at exactly this position of the missing PPs (naturally since 7+13=20 and does not belong on this row). What this points out is that the horizontal PP set pattern actually does follow the strict prime number sequence but for those PP sets that do NOT add up to the even number that that row represents, a blank naturally occurs there, causing the jumps in the strict prime number sequence. This is valuable information in confirming the horizontal, vertical and diagonal patterns of the PP sets are properly in place and accounted for.

Remembering that: 

• all primes above 2 are odd;

• that the double of every prime gives an even number not divisible by 4 [the numbers from the excluded evens of the 1^st Diagonal on Brooks (Base) Square];

• all even numbers are the product of two primes (Euler’s Goldbach Conjecture);

• those even #s that are the result of doubling a prime are always separated by a difference of a multiple of 4 from previous even #s so formed;

• the even numbers in between being those that are divisible by 4 [the numbers from the included evens of the 1^st Diagonal on Brooks (Base) Square] and even numbers not divisible by 4 [the numbers from the excluded evens of the 1^st Diagonal on Brooks (Base) Square];

let’s proceed to an example of a series of primes providing both a visual and mathematical solution to the solution of the conjecture.

• take prime numbers: 1601-1607-1609-1613-1621-1627-1637-1657-1663-1667-1669-1697-1699-1709;

• their gaps run: 6-2-4-6-2-6-10-20-6-4-2-24-4-2-10, respectively;

• we know that their doubles=even #s that are NOT divisible by 4 (excluded evens): 3202-3214-3218-3226-3238-3242-3254-3274-3314-3326-3334-3338-3386-3394-3398-3418;

• in between these double prime even #s are 92 other even #s (included and excluded);

• on the PTOP, the # of diagonals of PP sets increase by 1, starting with 3+3;

• prime # 1601 is ∑#PP = 251, i.e. there are 251 PP sets forming the 1601 diagonal;

• as each diagonal member occupies one prime column across, there are also 251 unique,

 possible PP sets going across the PTOP whose 2^nd PP set # is 1601;

• the last PP set, consisting of 1601 + 1601 equals 3202;

• the other 250 PP sets occupy 1PP set/even # less than 3202;

• the ∑#PP starts at column 3 PP set: 3 + 1601 = 1604;

• therefore, the ∑#PP 251 runs from 3=1601=1604 to 1601+1601=3202;

• between 1604 and 3202 there are 1598 total numbers, divided by 2 = 799 even #s;

• add 1 to include the 1^st even, 799 + 1 = 800 even #s;

• ∑#PP occupies 251 of 800, or 251/800, possible PP sets that sum up to those even #s;

• now, do the same distribution analysis for each subsequent prime: 1607-1609-1613-...;
• 1607+1607=3214 and 3214-(3+1607)=1604 and 1604/2=802 and 802+1=803 for 252/803;
• 1609+1609=3218 and 3218-(3+1609)=1606 and 1606/2=803 and 803=1=804 for 253/804;
• 1613+1613=3226 and 3226-(3+1613)=1610 and 1610/2=805 and 805+1=806 for 254/806;
• simplifying the calculation;
• (1619x2)-(3+1619)=1616 and (1616/2)+1=809 or (1619-1)/2=809 to give 255/809;
• (1621-1)/2=810 for 256/810;
• (1627-1)/2=813 for 257/813;
• (1637-1)/2=818 for 258/818;
• (1657-1)/2=828 for 259/828;
• (1663-1)/2=831 for 260/831
• (1667-1)/2=833 for 261/833;
• (1669-1)/2=834 for 262/834;
• (1693-1)/2=846 for 263/846;
• (1697-1)/2=848 for 264/848;
• (1699-1)/2=849 for 265/849;
• (1709-1)/2=854 for 266/854;

• knowing that every PP set is unique and appears only one time on the PTOP, and;

• knowing that every PP set has the 1^st # ≤ to the 2^nd # of the set, then;

• it would appear that the combined # of unique PP sets, as members of the ∑#PP, overlaps every possible even # to such an extent that it is statistically improbable ... impossible ... for any even #, greater than or equal to 4, NOT to be composed of at least one PP set;
• the ratio of 251/800 tells us that of the 800 possible even #s between 1604-3202, 251 PP sets, each of who's sums equal one of the the 800 even #s, 31+% are immediately covered by the 1601 diagonal of PP sets ... and, of course, the 1^st (1604) and last (3202) are known to be covered;

• when we move down to the next number, 1607, and apply the same distribution analysis, the 252/803 ratio shows a similar 31+% coverage for even #s 1607-3214;

• we see that most of the 803 even #s overlap and include the same even #s from the first analysis (1601);

• and so on, down the line;

• in fact, the amount of overlap of the PP sets covering the even #s is so vast that it is statistically improbably ... impossible ... for all the even #s NOT to be covered;

• not to forget, there are a number of PP diagonal sets both before prime 1601 and after 1709 that also overlap the even # sequences we have been looking at here but have not be included in the distribution analysis ... further strengthening the case presented;

• in order to find a hole/holes in the PP set coverage of the even #s, the gap between subsequent prime numbers would have to be so large, AND, such gaps would have to occur in succession over multiple neighboring primes to break through the extensive overlapping of PP sets that ensures that Euler’s  “strong” form of the Goldbach Conjecture is satisfied;

• as of early 2010, no such violation in over 4x10¹⁴ tested numbers have been found, the largest confirmed gap between primes is 1476 ... and is rare ... and Bertrand’s Postulate, proved in 1850 by Chebyshev, that between every number (>3) and its double-2, there exists a prime, highly suggests ... along with Periodic Table of Primes (PTOP) ... that the solution to the conjecture is no longer an open question.