The "Vertical" (generalization of) the binary Goldbach Conjecture (VBGC)

A very brief presentation of VBGC

"VBGC" is the abbreviation for "Vertical" generalization of the binary Goldbach conjecture (BGC) launched by dr. Andrei-Lucian Drăgoi (URL2) Because it is very simple and elegant (thus has an inner beauty that few have observed), VBGC is relatively easy to be formulated (see next).

Indexing the set of primes as p(1)(=2), p(2)(=3), p(3)=5, ....p(x)(the x-th prime), ... p(y)(the y-th prime), ... (with x and y being positive integers > 0 and p being treated as a bijective „indexing” function which connects each x index with the x-th prime p(x) and vice versa) and noting the prime-indexed primes (of any order of p-on-p iteration) as a_P( x)=p(p(...p(x))) (with an integer a>=0 counting the number of p-on-p interations) and b_P( y)=p(p(...p(y))) (with an integer b>=0 counting the number of p-on-p interations) (with a_P( x) and b_P( y) being actually the generalization of super-primes, with p “indexing” function (PIF) being treated as an iterated function applied on itself as p-on-p-on-p-on…=p(p(p(..)))), the “analytic” variant of VBGC simply states that:
“all evens 2m larger than a specific finite integer limit 2*f(a,b) can be written as a sum of at least one pair of distinct prime-indexed primes a_P( x) & b_P( y), with these experimentally-calculated finite integer limits: f(0,0)=3, f(1,0)=3, f(2, 0) = 2 564, f(1,1) = 40 306, f(3,0) = 125 771, f(2,1) = 1 765 126, f(4,0) = 6 204 163, f(3,1) = 32 050 472, f(2,2) = 161 352 166, f(5,0) = 260 535 479, f(4,1) =?(finite), f(3,2) =?(finite), f(3,3) =?(finite) . . .”
The f(a,b) limits have a very interesting exponential-like distribution which inspired another two variants of VBGC extending the estimations of f(a,b) limits far beyond the experimentally tested limit of 2m=10^10: for these two additional variants of VBGC, see the next sections of this Vixrapedia page.
Each pair of integers (a,b) plus its assigned integer limit f(a,b) defines a distinct subconjecture named here VBGC(a,b), with the (non-trivial) binary Goldbach conjecture (BGC) being only the first case VBGC(0,0) with L(0,0)=3. The subconjectures VBGC(a>0, and b>0) are much more elegant than BGC because they imply much fewer Goldbach partitions thus are much stricter/stronger than BGC (having much narrower Goldbach comets) and also suggest/indicate that BGC is very probably true as the simplest subconjecture of VBGC. That is why I’ve called VBGC a “meta-conjecture” defined here as class/set containing an infinite number of conjectures stronger/stricter than BGC and indexed in a 2D array/matrix as VBGC(a,b): VBGC can be actually regarded as a work of metamathematics (MM) or at the bridge between mathematics and MM, because it states that the distribution of primes iteratively (and infinitely!) applied on itself (as also described by the iterated PIF) maintains this “Goldbach property” that all evens above a finite integer limit 2*f(a,b) can be written as the sum of at least one pair or distinct primes a_P( x) & b_P( y).
VBGC is a very important generalization of BGC (independently verified by many OEIS reviewers) and that is the reason why it was accepted by one of the largest online mathematical encyclopedia -- OEIS (containing various integer sequences which Wikipedia frequently cites in its technical articles). VBGC and its results were independently verified (with distinct software, up to 2m=10^10 limit) and accepted in the OEIS encyclopedia as the main reference of the sequences A282251 and A316460 in 2017 and 2018 respectively.
See my VBGC ArXiv preprints here: URLv1, URLv2 URL2
OTHER ESSENTIAL INFORMATION ON VBGC:

1. VBGC was also recently cited in an article called "Os números primos de Ishango" by the Portuguese mathematician and researcher dr. Carla Santos PhD. See URLs: URL1 and URL2
2. My VBGC article was also cited by the well-known Chinese mathematician Zhi-Wei Sun (URL2 URL3) as a reference of an integer sequence he has submitted to OEIS in 2014 (and then updated it by linking VBGC as a main reference): A218829
3. I've recently published a free book (in A5 format) on VBGC called "The "Vertical" Generalization of Goldbach's Conjecture (VBGC) - An Infinite Class of Conjectures Stronger than Goldbach's" which is available for download in full text at these URLs: URL1(RG), URL2(Academia)
4. The main reference of VBGC: Dragoi, Andrei-Lucian (2017). "The “Vertical” Generalization of the Binary Goldbach’s Conjecture as Applied on “Iterative” Primes with (Recursive) Prime Indexes (i-primeths)" (original-research-article). Journal of Advances in Mathematics and Computer Science, ISSN: 2456-9968, ISSN: 2231-0851 (Past),Vol.: 25, Issue.: 2. URLs:
   1. http://www.sciencedomain.org/abstract/21625
   2. http://www.journalrepository.org/media/journals/JAMCS_69/2017/Oct/Andrei2522017JAMCS36895.pdf
   3. https://www.researchgate.net/publication/320740914

General facts on VBGC

The "Vertical" generalization of the binary Goldbach conjecture (BGC) launched by dr. Andrei-Lucian Drăgoi (who abbreviated it as "VBGC") is essentially a meta-conjecture, because it states a new class of conjectures containing a potentially infinite number of conjectures stronger/stricter than BGC: this is the new sense of the "meta-conjecture" concept used here, so that to be differentiated from the standard/classical "metaconjecture" definition.

The main website of Dr. Dragoi is available at this URL.

A short latex paper on VBGC was also launched in 11.02.2020 ("(VBGC - 2020 latex version in short - 11.02.2020 - 5 A4 pages without references) A "Vertical" Generalization of the binary Goldbach's Conjecture (VBGC) as applied on primes with prime indexes of any order i ("i-primes") (VBGC: A new set/class containing an infinite number of Goldbach-like conjectures stronger than the BGC)") (with DOI: 10.13140/RG.2.2.29775.02725) and is available at these URLs: URL-RG and URL-Academia

VBGC was independently verified by OEIS reviewers and initially approved by OEIS in Feb 10 2017, as the main reference of the sequence A282251. See also URL. See also URL with keyword "Andrei-Lucian Drăgoi".

VBGC was independently re-verified by OEIS reviewers and again re-approved by OEIS in July 04 2018, as the main reference of the sequence A316460. See also URL. See also URL with keyword "Andrei-Lucian Drăgoi".

VBGC is also promoted by Google Books at URL.

VBGC is also the object of a polemic between the author (dr. Dragoi) and Mr. Bernard Peter Gore on Quora: see URL

VBGC may be also cited as "Goldbach-Dragoi binary meta-conjecture" (and abbreviated as GDBMC or DBMC), so that to be differentiated from the "Vertical" (generalization of) the ternary Goldbach Conjecture (VTGC)" (or "Goldbach-Dragoi ternary meta-conjecture" [abbreviated as GDTMC or DTMC]) which is stil "under construction" by the same author.

VBGC uses the following essential notions:

1. the indexed notation of positive prime numbers, which is essentially an indexing/numbering function of (positive) primes aka "P" (P₁=2, P₂=3, P₃=5,... P_n..., with P_n being the n-th prime number of the primes sequence A000040) or (alternatively) P(1)=2, P(2)=3, P(3)=5,...P(n)=P_n..., with additional notation of P-on-P recursive function P(P(n))=P_P(n)=P_Pn.
2. the binary Goldbach Conjecture (abbreviated as "BGC" in this article) and, more specifically, the non-trivial BCG (abbreviated as "ntBGC") which only considers the Goldbach partitions composed from distinct (non-identical) positive primes.
3. the super-prime numbers subsequence A006450(also known as "higher-order primes" or "prime-indexed primes"), which are the subsequence of prime numbers P_n with integer index "n" being also a positive prime number: this is equivalent to prime numbers of form P(P(n))=P_P(n)=P_Pn.
4. the generalized concept of a positive "i-primeth" (an "order-i primeth" with i being an integer index i≥0, as proposed by dr. Andrei-Lucian Drăgoi) defined and noted as: ⁰P_n=P_n (a 0-primeth with index n, which is equivalent to a positive prime number with index n, with zero P-on-P iterations), ¹P_n=P(P_n) (a 1-primeth with index n, which is equivalent with a super-prime with index n and which implies one single P-on-P iteration), ²P_n=P(P(P_n)) (a 2-primeth with index n, which implies two P-on-P iterations),... ⁱP_n=P(P...(P_n)) (an i-primeth with index n, which implies a number of "i" P-on-P iterations). Note. This type of inverse exponential notation (ⁱP_n=P(P...(P_n), implying a number of "i" P-on-P iterations) is used to strictly number the number of P-on-P iterations, which is an alternative to the normal (exponential) notation of iterated functions (Pⁱ_n=P(P...(P_n), implying a number of just "i-1" P-on-P iterations corresponding to P applied on "n" and itself "i" times in total), an alternative which avoids some possible confusions with an exponential value of a function, which is noted (P_n)ⁱ=[P(n)]ⁱ)

Essentially, VBGC states that: "Any positive even integer larger than a specific finite (positive) limit f(a,b) can be written as the sum of two distinct i-primeths ^aP_x and ^bP_y."

OR (alternative formulation)
"Any positive integer larger than a specific finite (positive) limit f(a,b)/2 can be written as the arithmetic average of two distinct i-primeths ^aP_x and ^bP_y."

Based on VBGC, dr. Dragoi also proposes the generalized concept of Goldbach i-primeths partitions (or briefly "Goldbach i-partitions (GIP)") with composed order (a,b) (briefly noted GIP(a,b)) of any even (positive) integer 2m, counting all the ways in which (that) 2m can be written as the sum of two distinct i-primeths ^aP_x and ^bP_y. To be compared with the special case generically called "Goldbach partitions".

VBGC also states that the (apparently unpredictable) non-uniform distribution of primes (D_P) is such as no matter how many times D_P is iteratively applied on the set of naturals ₦ and/or on itself (the set of primes), the (positive) limit above which all (positive) integers can be always written as a sum of two distinct i-primeths is always finite. From this point of view and based on this special property of D_P, the set of primes (including all i-primeths subsets) may be regarded a quasi-random fractal with a quasi-random fractal distribution D_P. Given this very special (almost "providential") property of D_P (claimed by VBGC and its author), VBGC may bring a new light on Riemann hypothesis, Riemann zeta function and their generalized forms: Generalized Riemann hypothesis and Grand Riemann hypothesis (all included, along with BGC, in Hilbert's eighth problem posed in 1900).

VBGC has multiple variants (see next sections):

The "inductive" VBGC (iVBGC)

iVBGC states that:
"All even positive integers 2m≥2f_x1(a,b) AND (also) 2m≥2f_x2(a,b) can be written as the sum of at least one pair of DISTINCT odd positive i-primeths ^aP_x>^bP_y, WITH:
f_x1(a,b)=2^{(a+1)(b+1)(a+b+2)} if (a=b=0)
f_x1(a,b)=2^{[(a+1)(b+1)(a+b+3)/a]-a} if [(a=b) AND (a>0)]
f_x1(a,b)=2^{(a+1)(b+1)(a+b+2)-(a+b-2)} if {(a≠b) AND [(a>0) OR (b>0)]}
AND
f_x2(a,b)=2^{(a+1)(b+1)(a+b+2)} if (a=b=0)
f_x2(a,b)=2^{[(a+1)(b+1)(a+b+3)/a]-2a} if [(a=b) AND (a>0)]
f_x2(a,b)=2^{(a+1)(b+1)(a+b+2)-(a+b-2)} if {[(a≠b) AND [(a>0) OR (b>0)]}"

• Important note. iVBGC is essentially a meta-conjecture, as each (a,b) pair generates a pair of finite function values f_x1(a,b) and f_x2(a,b) which corresponds to a pair of distinct (sub)conjectures which can be indexed as iVBGC₁(a,b) and iVBGC₂(a,b) respectively.

The secondary "inductive" VBGC (siVBGC)

siVBGC is actually "appendix" of iVBGC and states that:
"All even positive integers 2m≥2floor(f_y1(a)) (with f_y1(a)=e^4a) can be written as the sum of at least one pair of DISTINCT positive odd i-primeths ^aP_x>⁰P_y."

• Important note. siVBGC is essentially a meta-conjecture, as each (a,0) pair generates a finite function value f_y1(a,0) which corresponds to a distinct (sub)conjecture which can be indexed as siVBGC(a,0).

siVBGC has also a general(g) form abbreviated "gsiVBGC" (which gsiVBGC is however much "weaker" than iVBGC when comparing the finite inferior integer limits f_x1,2(a,b) of iVBGC versus f_y1,2(a,b) of gsiVBGC for their validity) which states that:
"All even positive integers 2m≥2floor(f_y2(a,b)) (with f_y2(a,b)=e^4(a+b)) can be written as the sum of at least one pair of DISTINCT positive odd i-primeths ^aP_x>^bP_y."

• Important note. gsiVBGC is essentially a meta-conjecture, as each (a,b) pair generates a finite function value f_y2(a,b) which corresponds to a distinct (sub)conjecture which can be indexed as gsiVBGC(a,b).

The "analytical" variant of VBGC (aVBGC)

aVBGC in fact the "original" VBGC (from which iVBGC was "induced") and states that:
"All even positive integers 2m>2f(a,b) (with f(a,b)=f(b,a) co-stated to be a finite positive integer for any finite a and b) can be written as the sum of at least one pair of DISTINCT positive odd i-primeths ^aP_x>^bP_y, WITH: f(0,0)=3, f(1,0)=3, f(2,0)=2'564, f(1,1)=40'306, f(3,0)=125'771, f(2,1)=1'765'126, f(4,0)=6'204'163, f(3,1)=32'050'472, f(2,2)=161'352'166, f(5,0)=260'535'479, f(3,2)=?, f(4,1)=? (expected to be smaller than f(3,2)=?), f(3,3)=?" (all function values marked with "?" necessitate more powerful computers and software to be found and verified at least up to 10¹¹)

• Important note. aVBGC is essentially a meta-conjecture, as each (a,b) pair generates a finite function value f(a,b) which corresponds to a distinct (sub)conjecture which can be indexed as aVBGC(a,b).

Comments (including the history of VBGC)

1. VBGC was discovered in 2007 and perfected until 2017 by using the matrices Goldbach index-partitions. Initially, a part of aVBGC(a,b) (sub)conjectures were tested by dr. Andrei-Lucian Dragoi up to 2m=10⁹, using a software built in Visual Basic 6 (VB6) and Visual Basic for Applications (VBA). Then, all variants of VBGC (iVBGC, siVBGC and aVBGC) were tested up to 2m=10¹⁰, in collaboration with Mr. George Anescu (who is also listed on URL2), a mathematician friend who created the Visual C++ variant of the testing software which can be freely downloaded from this public Google Drive link.
2. Each VBGC using a specific f(a,b) is actually a separate (sub)conjecture that is noted VBGC(a,b). Each VBGC(a,b) (sub)conjecture (other than BGC) has an associated Goldbach-like comet which is much more narrow than the "classical" Goldbach comet (associated with VBGC(0,0)).
3. BGC is represented by the special case VBGC(0,0).
4. VBGC(1,0) has its f(1,0) equal to f(0,0) (of BGC) which makes VBGC(1,0) "stronger"/stricter and more elegant than BGC, as it remarkably has the same even exceptions as the non-trivial BCG (just the even positive integers 2,4 and 6, as 0 is a trivial even exception in this case), but offers a significantly smaller number of Goldbach partitions for each 2m than BGC does. I have tried to publish VBGC(1,0) together with its very small sequence of exceptions (the evens 2,4 and 6) on OEIS as A316297 but I was treated abusively and unprofessionally (I was even offended by one moderator which showed such a narrow and superficial review capabilities and also arrogant and unethical behavior): this dispute can be found at this URL (which contains many pages accessible from the "older changes" tab). The contradiction between me and Mr. Neil Sloane (the founder of OEIS) can be found in pdf format at this URL
5. The exceptions of VBGC(1,1) (see the indexed exceptions list, also called "b-file", at this URL) were also approved on OEIS as A316460: see also the open review history of A316460 sequence at this URL. I've also had a controversy with Mr. Neil Sloane (the founder of OEIS) which abusively decided to block me from submitting new sequences on OEIS for 12 months: this dispute can be found at this URL
6. I have initially proposed the meta-sequence of f(a,b) values (stated by aVBGC) to OEIS, as A281929: although VBGC was also tested and confirmed by OEIS moderators at that time, it was rejected with the main argument that it was "too ambitious" and that OEIS cannot accept "meta-sequences" (although this meta-sequence had the potential to be the most interesting and elegant meta-sequence ever published on BGC); the dispute between me and OEIS stuff on A281929 (which proves the rigid approach practiced by OEIS) can be found at this URL (multiple pages accessible in the tab "older changes") and at this URL in pdf format. In the end, at the proposal of OEIS moderators, the initially intended ("maximally" elegant) meta-sequence was replaced with (the more "modest") VBGC(2,0) even exceptions sequence, which was also reviewed, re-verified with Mathematica software (together with all variants of VBGC: iVBGC, siVBGC and aVBGC) and approved on OEIS as sequence A282251. The review history of A282251 can be also found at this URL (which contains many pages accessible from the "older changes" tab) and at this URL (in pdf format).
7. I have also send my VBGC article to Wolfram's MathWorld site as mathematical news (and web article subject or section in their Goldbach conjecture page) both in 2017 and 2018 but never received an answer, not even a feedback email to confirm that they have received the news(/article/section proposal) but rejected it for any specific reason.
8. The irrational (and non-argumented) censorship of Wikipedia moderators is also almost proverbial: some disputes on VBGC can be read at these URLs: URL1a, URL1b, URL1c and URL2.
9. On June 29th 2018, I have also initiated a discussion thread on Quora about VBGC, which also started a dispute on Quora: see URL (I shall periodically upgrade the status of this fact in the future).
10. On July 27th 2018, I have also sent VBGC to Clay Mathematics Institute (CMI) with the proposal that VBGC (and finding its formal validation/invalidation proof) to be included on the list of the Millennium Prize Problems, with no feedback from CMI until present (I shall periodically upgrade the status of this fact in the future).
11. I have also sent VBGC in July 2018 to other well-known mathematicians, with no feedback from them until present (I shall periodically upgrade the status of this fact in the future).
12. As an additional note (see VBGC reference), VBGC can be considered an indirect „proof” of BGC, because VBGC essentially states/conjectures an infinite set of finite values f(a, b) which indicates that BGC (equivalent to VBGC (0,0)) is very probably true, because f(0,0) is only a special case (the first one) of this (conjectured) infinite set. In other (more plastic) words, BGC is just a "tree" in the plausibly infinite VBGC "wood", which VBGC is a spectacular quasi-fractal property of primes distribution (Dp) when applied iteratively on itself (and holding VBGC).
13. VBGC can be also used in N-dimensional spaces analysis, which analysis is also interconnected with string theory and M-Theory (see the main VBGC article).
14. VBGC can also be used for significantly optimizing/speeding the various empirical BGC verification algorithms up to integer limits much higher than ~10¹⁸, which is the highest tested limit for BGC validity until present (as explained in the main reference VBGC article).
15. A plastic conclusion. As all (positive) integers above a certain (finite) limit can always be written as the arithmetic average of two distinct i-primeths (the alternative general formulation of VBGC), all (positive) i-primeths can be metaphorically regarded as actually organized similarly to the DNA "double helix" of living organisms (as they can be also metaphorically resembled to a "genome" of all positive integers) around all (positive) integers (which play the role of a symmetry axis, a kind of "backbone" of this "DNA" composed from i-primeths): to picture this visual metaphor, for each (a,b) pair, one can count the number of GIPs(a,b) of all (positive) integers (integers that can be written as the arithmetic average of two distinct i-primeths ^aP_x and ^bP_y) and create a 3D image in which the axis of positive integers is intersected (in its "integer" points) by a regular polygon with a number of sides equal to the number of GIPs(a,b) of that (positive) "integer" (point); VBGC essentially states/claims that this "double helix"-like structure of i-primeths has no discontinuities over a certain positive (finite) limit f(a,b)/2.
16. Dr. Andrei Lucian Dragoi has multiple user accounts/pages on:

• Research Gate: https://www.researchgate.net/profile/Andrei_Lucian_Dragoi2
• Academia.edu: https://univermed-cdgm.academia.edu/AndreiLucianDragoi
• Vixra.org: http://vixra.org/author/andrei_lucian_dragoi
• GSJournal.net: http://gsjournal.net/Science-Journals-Papers/Author/1713/Andrei-Lucian,%20Dragoi
• OEIS: https://oeis.org/wiki/User:Andrei-Lucian_Dragoi

References

Dragoi, Andrei-Lucian (2020). "(VBGC - 2020 latex version in short - 11.02.2020 - 5 A4 pages without references) A "Vertical" Generalization of the binary Goldbach's Conjecture (VBGC) as applied on primes with prime indexes of any order i ("i-primes") (VBGC: A new set/class containing an infinite number of Goldbach-like conjectures stronger than the BGC)" (DOI: """10.13140/RG.2.2.29775.02725"""). Available at these URLs: "URL-RG" and "URL-Academia"

Dragoi, Andrei-Lucian (2017). "The “Vertical” Generalization of the Binary Goldbach’s Conjecture as Applied on “Iterative” Primes with (Recursive) Prime Indexes (i-primeths)" (original-research-article). Journal of Advances in Mathematics and Computer Science, ISSN: 2456-9968, ISSN: 2231-0851 (Past),Vol.: 25, Issue.: 2. URLs:

1. http://www.sciencedomain.org/abstract/21625
2. http://www.journalrepository.org/media/journals/JAMCS_69/2017/Oct/Andrei2522017JAMCS36895.pdf
3. https://www.researchgate.net/publication/320740914

• This (freely available) full-text article also contains: (1) other Goldbach-like conjectures (proposed by other authors) stronger/stricter than BGC; (2) dr. Dragoi's proposed general classification of all possible Goldbach-like conjectures in two major types: type A and type B (see the article for the exact definitions of these A and B major types).

The list of Vixrapedia pages created by dr. Andrei-Lucian Drăgoi (in the chronological order of release)

1. The "Vertical" (generalization of) the binary Goldbach Conjecture (VBGC) (versiune în limba engleză/English version only)
2. Robert Klenck (Romanian violinist and violin professor) (versiune în limba engleză)
3. Robert Klenck (violonist și profesor de vioară) (versiune în limba română)
4. The Doctors Orchestra "Ermil Nichifor" (Bucharest, Romania) (versiune în limba engleză)
5. Orchestra Medicilor "dr. Ermil Nichifor" (București, România)] (versiune în limba română)