A "Vertical" Generalization of the binary Goldbach's Conjecture as applied on primes with prime indexes of any order i (i-primes)
AA Matricial Aspect of Goldbach’s Conjecture
Andrei-Lucian Dr˘agoi Abstract
This article is a survey based on our earlier paper (“The “Vertical” Generalization of the Binary Goldbach’sConjecture as Applied on “Iterative” Primes with (Recursive) Prime Indexes (i-primeths)” [11] ), a paper inwhich we have proposed a new generalization of the binary/“strong” Goldbach’s Conjecture ( GC ) briefly called“the Vertical Goldbach’s Conjecture” ( VGC ), which is essentially a metaconjecture, as VGC states an infinitenumber of Goldbach-like conjectures stronger than GC, which all apply on “iterative” primes with recursiveprime indexes (named “i-primes”). VGC was discovered by the author of this paper in 2007, after which it wasimproved and extended (by computational verifications) until the present (2019). VGC distinguishes as a veryimportant “metaconjecture” of primes, because it states a new class containing an infinite number of conjecturesstronger/stricter than GC. VGC has great potential importance in the optimization of the GC experimentalverification (including other possible theoretical and practical applications in mathematics and physics). VGCcan be also regarded as a very special self-similar property of the primes distribution. This present survey containssome new results on VGC.
Keywords : primes with recursive prime indexes (i-primes); the binary/strong Goldbach Conjecture ( GC );the Vertical (binary/strong) Goldbach Conjecture ( VGC ), metaconjecture.
Mathematical subject classification codes : 11N05 (distribution of primes), 11P32 (Goldbach-type theo-rems; other additive questions involving primes), 11Y16 (algorithms; complexity); ***
Introduction
This paper proposes the generalization of the binary (aka “strong”) Goldbach’s Conjecture (
GC) [22] , brieflycalled “the Vertical (binary) Goldbach’s Conjecture” (
VGC ), which is essentially a meta-conjecture, as VGC statesan infinite number of Goldbach-like conjectures stronger/stricter than GC, which all apply on “iterative” primeswith recursive prime indexes named “i-primes” in this paper. ***
Given the simplified notation (used in this article) p x = p ( x ) ≥ x ∈ N ∗ = { , , , ... } ) as the x-thprime from the infinite countable primes set P = { p (= 2) , p (= 3) , p (= 5) , ...p x , ... } , the “i-prime” concept isthe generalization with iteration order i ∈ N = { , , , , ... } of the known “prime-indexed primes” (alias “super-primes”) as a subset of (simple or recursive) primes with (also) prime indexes, with i p x = p i +1 ( x ) = p ( p ( p... ( x ))) (cid:54) =[ p ( x )] i +1 (the primes indexing bijective function “p” applied (i+1)-times on any x ∈ N ∗ , which implies a number ofjust i p-on-p iterations) being the x-th i-prime, with iteration order i ∈ N , as noted in this paper. In this notation,simple primes are defined as 0-primes so that p x = p x = p ( x ) (and P = P ): p x = p p x = p ( x ) = p ( p ( x )) arecalled “1-primes” (cid:0) P = (cid:8) p , p , ... (cid:9) ⊂ P (cid:1) (with just one p-on-p iteration), p x = p p px = p ( x ) = p ( p ( p ( x )))are called “2-primes” (cid:0) P = (cid:8) p , p , ... (cid:9) ⊂ P ⊂ P (cid:1) (with just two p-on-p iterations) and so on (cid:0) i P ⊂ P (cid:1) .There are a number of (relative recently discovered) Goldbach-like conjectures ( GLCs ) stronger than GC: these stronger GLCs (including VGC, defined as a collection of an infinite number of GLCs) are tools that caninspire new strategies of finding a formal proof for GC, as I shall try to argue in this paper . VGC distinguishes as a very important metaconjecture of primes (with potential importance in the optimizationof the GC experimental verification and other possible useful theoretical and practical applications in mathematics[including cryptography and fractals] and physics [including crystallography and M-Theory]), and a very specialself-similar property of the primes distribution ( PD ).GC is specifically reformulated by the author of this article as a special property of PD, in order to emphasizethe importance of PD study [10, 13, 14, 21, 26] , which PD has multiple interesting fractal patterns . E-mail: [email protected] a r X i v : . [ m a t h . G M ] O c t he non-trivial GC ( ntGC ) variant (excluding the trivial cases for identical prime pairs p x = p y ) essentiallystates that: PD is sufficiently dense and (sufficiently) uniform, so that any natural even number n ( with n > can be written as the sum of at least one Goldbach partition ( GP ) of distinct primes p x > p y , so that 2 n = p x + p y . *** ntGC redefinition . Let us define the set of GP matrices M = (5 ,
3) (with 2 ∗ M = (7 ,
3) (with 2 ∗ M = (7 ,
5) (with2 ∗ M = (11 ,
3) (with 2 ∗ M = (cid:18) , , (cid:19) (with 2 ∗ M n = P x , P y ( < P x ) P x , P y ( < P x ) ... , ... , with p x + p y = 2 np x + p y = 2 n... , n > p x (cid:54) = p y ) plus the GP-non-redundancy condition in M n (which actually implies the non-triviality one) ( p x > n ) ⇒ ( p y < n )( ⇒ p x (cid:54) = p y ), which is an additional condition imposed to eliminate redundant lines of M n (which maycontain the same elements as another “mirror”-line, but in inversed order): the ( p x > n ) condition also anticipatesthe ntGC verification algorithm proposed by VGC, which algorithm starts from the p x (?) (which is the primeclosest to, but smaller than 2 n ), scans all primes p k (cid:0) ≤ p x (?) (cid:1) downwards to p = 3 and verifies the primality ofthe differences d k (= 2 n − p k ).Based on M n (with entries in P = P for any n >
3) and defining the empty matrix M ∅ (with zero lines, thusalso zero columns), ntGC can be restated as: for any positive integer n > , M n (cid:54) = M ∅ ( ntGC ) *The “Goldbach-like Conjecture (GLC)” definition . A GLC is defined in this paper as the combinationof GC plus any additional conjectured property of M n (other that M n (cid:54) = M ∅ ) for any positive integer n > L (with L ≥ GLCs classification . GLCs may be classified in two major classes using a double criterion such as:1.
Type A GLCs (A-GLCs) are those GLCs that claim: ( ) Not only that all M n (cid:54) = M ∅ for any n > L butalso ( ) any other non-trivial accessory property/properties common for all M n ( (cid:54) = M ∅ ) . A specific A-GLCis considered “authentic” if the other non-trivial accessory property/properties common for all M n ( (cid:54) = M ∅ ) (claimed by that A-GLC) isn’t/aren’t a consequence of the 1 st claim (of the same A-GLC). Authentic (at leastconjectured as such) A-GLCs are (have the potential to be) “stronger”/stricter than ntGC as they essentiallyclaim “more” than ntGC does. Type B GLCs (B-GLCs) are those GLCs that claim: no matter if all M n (cid:54) = M ∅ or just some M n (cid:54) = M ∅ for n > L , all those M n that are yet non- M ∅ (for n > L ) have (an)other non-trivial accessory property/propertiescommon for all M n (cid:54) = M ∅ (for n > L ) . A specific B-GLC is considered authentic if the other non-trivialaccessory property/properties common for all M n ( (cid:54) = M ∅ ) (claimed by that B-GLC for n > L ) isn’t/aren’t aconsequence of the fact that some M n (cid:54) = M ∅ for n > L . Authentic (at least conjectured as such) B-GLCs are“neutral” to ntGC (uncertainly “stronger” or “weaker” conjectures) as they claim “more” but also “less”than ntGC does (although they may be globally weaker and easier to be formally proved than ntGC). * Other variants of the (generic) Goldbach Conjecture ( GC ) and related conjectures include:1. “Any odd integer n > ”. This is the (weak) TernaryGoldbach’s conjecture ( TGC ), which was rigorously proved by Harald Helfgott in 2013 [15, 16] (a complexproof that is generally accepted as valid until present), so that TGC is already considered a proved theorem,and no longer just a conjecture. 2. “Any integer n >
17 can be written as the the sum of exactly 3 distinct primes ”. This is cited as “Conjecture3.2” by Pakianathan and Winfree in their article and is a conjecture stronger than TGC, but weaker thanntGC.3. “Any odd integer n > ”. This is Lemoine’s conjecture ( LC ) [19] which is sometimes erroneously attributed to Levy H. whopondered LC in 1963 [20] . LC is stronger than TGC, but weaker than ntGC. LC has also an extensionformulated by Kiltinen J. and Young P. (alias the ”refined Lemoine conjecture” [18] ), which is stronger thanLC, but weaker than ntGC and won’t be discussed in this article (as this paper mainly focuses on thoseGLCs stronger than ntGC).There are also a number of (relative recently proposed) GLCs stronger than ntGC (and implicitly strongerthan TGC), that can also be synthesized using the M n definition and A/B GLCs classification: these strongerGLCs (as VGC also is) are tools that can inspire new strategies of finding a formal proof for ntGC (but alsooptimizing the algorithms of ntGC empirical verification up to much higher limits than in the present), as we shalltry to argue next. The Goldbach-Knjzek conjecture [GKC] [24] (which is stronger than ntGC) (slightly reformulatednoting with the even number with 2 n [not with n ], so that each ): “ For any even integer n > there is atleast one prime number p [so that] √ n < p ≤ n and q = 2 n − p is also prime [with n = p + q implicitly] ”.GKC can also be reformulated as: “ every even integer n > is the sum of at least one pair of primes withat least one prime in the semi-open interval (cid:0) √ n, n (cid:3) ”. GKC can be also formulated using M n such as:(a) Type A formulation variant : “For any even integer 2n > M n (cid:54) = M ∅ and M n contains at leastone line with one element p ∈ (cid:0) √ n, n (cid:3) .” (b) Type B (neutral) formulation variant : “For any even integer 2n >
4, those M n which are (cid:54) = M ∅ will contain at least one line with one element p ∈ (cid:0) √ n, n (cid:3) .” (c) A non-trivial GKC (ntGKC) version (which excludes the cases p = q = n ) is additionally proposedin this paper (with A/B formulations analogous to the GKC variants) and verified up to 2 n = 10 :“ every even integer n > is the sum of at least one pair of distinct primes with one prime p ∈ (cid:0) √ n, n (cid:1) ”.2. The Goldbach-Knjzek-Rivera conjecture [GKRC] [25] (which is also obviously stronger than ntGC,but also stronger than GKC for n ≥
64) (reformulated): “ ∀ even integer n > , there is at least one primenumber p [so that] √ n < p < √ n and q = 2 n − p is also prime [with n = p + q implicitly] ”. GKRCcan also be reformulated as: “ ∀ even integer n >
4, 2 n is the sum of at least one pair of primes with oneelement in the double-open interval (cid:0) √ n, √ n (cid:1) ”. GKRC can be formulated using M n such as:(a) Type A formulation variant : “For any even integer 2n > M n (cid:54) = M ∅ and M n contains at leastone line with one element p ∈ (cid:0) √ n, √ n (cid:1) .” (b) Type B (neutral) formulation variant : “For any even integer 2n >
4, those M n which are (cid:54) = M ∅ will contain at least one line with one element p ∈ (cid:0) √ n, √ n (cid:1) .” (c) A non-trivial GKRC (ntGKRC) version (which excludes the cases p = q = n ) is additionallyproposed in this paper (with A/B formulations analogous to the GKRC variants) and verified up to2 n = 10 : “ every even integer n > is the sum of at least one pair of distinct primes with one primein the open interval (cid:0) √ n, √ n (cid:1) ”.3. Noting with g ( n ) the number of M n lines for each 2 n in part (in any GLC) (identical to the standardfunction g ( n ) counting the number of non-redundant GPs for each 2 n tested in ntGC), any other GLC thatestablishes an additional superior limit of g ( n ) (like Woon’s GLC [29] ) can also be considered stronger thatntGC, because ntGC only suggests g ( n ) > n > n than the more selective Woon’s GLC does).3 The main metaconjecture proposed in this paper: The Vertical Goldbach(meta)conjecture (VGC) - The extension and generalization of ntGC asapplied on i-primes
Alternatively noting p x = p ( x ) (the x-th 0-prime, equivalent to the x-th prime in the indexed primes set), p x = p ( p ( x )) (the x-th 1-prime), p x = p ( p ( p ( x ))) (the x-th 2-prime), . . . i p x = p i +1 ( x ) (the x-th i-prime: notto be confused with the exponential [ p ( x )] i +1 (cid:54) = p i +1 ( x )), the (main) analytical variant of VGC ( aVGC ) statesthat:“ For any pair of finite positive integers ( a, b ) , with a ≥ b ≥ defining the (recursive) orders of an a-prime ( a p ) and a distinct b-prime b p (cid:0) a p (cid:54) = b p, but not necessarily a p > b p when a (cid:54) = b (cid:1) respectively, there is a finite positiveinteger limit L ( a,b ) = L ( b,a ) ≥ such as, for any (positive) integer n > L ( a,b ) there will always exist at least onepair of finite positive integer indexes ( x, y ) so that a p x (cid:54) = b p y and a p x + b p y = 2 n . ”More specifically, VGC states/predicts that: “the 2D matrix L containing all the L ( a,b ) limits (organized oncolumns indexed with “a” and lines indexed with “b”) has a finite positive integer value in any ( a, b ) position,without any catastrophic-like infinities, such as (experimentally verified values): L (0 , = 3 , L (1 , = 3 , L (2 , =2 564 , L (1 , = 40 306 , L (3 , = 125 771 , L (2 , = 1 765 126 , L (4 , = 6 204 163 , L (3 , = 32 050 472 , L (2 , =161 352 166 , L (5 , = 260 535 479 , L (4 , =?( f inite ), L (3 , =?( f inite ), L (3 , =?( f inite ) . . . “ L = (3) (3) (2 564) (125 771) (6 204 163) (260 535 479)(3) (40 306) (1 765 126) (32 050 472) (?) (?)(2 564) (1 765 126) (161 352 166) (?) (?) (?)(125 771) (32 050 472) (?) (?) (?) (?)(6 204 163) (?) (?) (?) (?) (?)(260 535 479) (?) (?) (?) (?) (?) *Important notes on aVGC: