Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs
DDivisibility in paired progressions, Goldbach’sconjecture, and the infinitude of prime pairs
Mario Ziller and John F. Morack
Abstract
We investigate progressions in the set of pairs of integers Z and define ageneralisation of the Jacobsthal function. For this function, we conjecture a specificupper bound and prove that this bound would be a sufficient condition for thetruth of the Goldbach conjecture, the infinitude of prime twins, and more generalof prime pairs with a fixed even difference. Henceforth, we denote the set of integral numbers by Z and the set of natural num-bers, i.e. positive integers, by N . P = { p i | i ∈ N } is the set of prime numbers with p =
2. As usual, we define the n th primorial number as the product of the first n primes: p n = ∏ ni = p i , n ∈ N . For similar objects of the specific context, we followthe notation of a recent paper [8].We investigate the set of pairs of integers Z = Z × Z and first define divisibil-ity of pairs with weak postulations. Afterwards, Jacobsthal’s function [5, 3] will begeneralised for the case of progressions of consecutive integer pairs using this conceptof divisibility. In the subsequent sections, we demonstrate the relationship betweenvarious unsolved problems in number theory, including Goldbach’s conjecture andthe twin prime conjecture, with a specific bound of the generalised Jacobsthalfunction. Divisibility of integer pairs
Considering Z as a canonical module over Z with ( a , b ) + ( c , d ) = ( a + c , b + d ) and k · ( a , b ) = ( k · a , k · b ) for all a , b , c , d , k ∈ Z , would imply strong requirements ofdivisibility: k | ( a , b ) if and only if there exists ( c , d ) ∈ Z with ( a , b ) = k · ( c , d ) . Thisdefinition is equivalent to: k | ( a , b ) if and only if k | a and k | b . In this paper instead, weonly use the weak divisibility as follows. Definition 1.1.
Divisor of a pair.
An integer k ∈ Z is a divisor of a pair ( a , b ) ∈ Z if k | a or k | b . We simply write k | ( a , b ) . 1 a r X i v : . [ m a t h . N T ] J un emark 1.1. k | ( a , b ) ⇒ k | a · b .We remark that k | a · b does not necessarily mean k | ( a , b ) . If k = k · k , k | a , and k | b then k (cid:45) b and k (cid:45) a may pertain.But prime numbers, on the other hand, may be characterised by this condition: p ∈ P ⇔ ( p | a · b ⇒ p | ( a , b ) ) .As a consequence of the last definition we declare coprimeness accordingly. Definition 1.2.
Coprime.
An integer k ∈ Z and a pair ( a , b ) ∈ Z are coprime, and we write k ⊥ ( a , b ) , if k ⊥ a and k ⊥ b . Remark 1.2. k ⊥ ( a , b ) ⇒ k (cid:45) a · b .As a conclusion, an integer k ∈ Z and a pair ( a , b ) ∈ Z are not coprime, and we write k (cid:54)⊥ ( a , b ) , if k (cid:54)⊥ a or k (cid:54)⊥ b . In other words, there exists an k ∗ ∈ Z with k ∗ | k and k ∗ | ( a , b ) . Again, k itself is not necessarily a divisor of ( a , b ) here. Jacobsthal function
The ordinary Jacobsthal function j ( n ) is known to be the smallest natural number m ,such that every sequence of m consecutive integers contains at least one integer co-prime to n [5, 3]. Definition 1.3.
Jacobsthal function.
For n ∈ N , the Jacobsthal function j ( n ) is defined as j ( n ) = min { m ∈ N | ∀ a ∈ Z ∃ q ∈ {
1, . . . , m } : a + q ⊥ n } . Remark 1.3.
This definition is equivalent to the formulation that j ( n ) is the greatestdifference m between two terms in the sequence of integers which are coprime to n . j ( n ) = max { m ∈ N | ∃ a ∈ Z : a ⊥ n ∧ a + m ⊥ n ∧∀ q ∈ {
1, . . . , m − } : a + q (cid:54)⊥ n } .In other words, ( j ( n ) − ) is the greatest length m ∗ = m − n .An analogous function will be defined below for sequences of integer pairs in placeof consecutive integers. Paired progressions
Progressions of consecutive pairs in Z can be defined in an intuitive way. There isa natural duality between a progression of consecutive pairs of integers and a pair ofprogressions of consecutive integers. Definition 1.4.
Consecutive pairs. ( a , b ) , ( a , b ) ∈ Z are called consecutive pairs if a = a + b = b + ( a , b ) is called successor of ( a , b ) . 2 efinition 1.5. Paired progression.
An ordered sequence of consecutive pairs { ( a i , b i ) ∈ Z } i = k is called a pairedprogression if ( a i + , b i + ) is a successor of ( a i , b i ) for i =
1, . . . , k − Remark 1.4.
For every paired progression { ( a i , b i ) ∈ Z } i = k as defined above, thereexists a pair ( a , b ) ∈ Z with ( a i , b i ) = ( a + i , b + i ) for i =
1, . . . , k . We use the notation (cid:104) a , b (cid:105) k for this paired progression and point to the fact that ( a , b ) itself is not memberof the progression. We now generalise Jacobsthal’s function on successive pairs of integers as its canonicalextension to paired progressions and apply weak divisibility at it.
Definition 2.1.
Paired Jacobsthal function.
Let n be a natural number. The paired Jacobsthal function j ( n ) is defined to be thesmallest natural number m , such that every paired progression (cid:104) a , b (cid:105) m of length m withan even difference of its pair elements contains at least one pair coprime to n . j ( n ) = min { m ∈ N | ∀ ( a , b ) ∈ Z with | ( b − a ) : ∃ ( x , y ) ∈ (cid:104) a , b (cid:105) m : n ⊥ ( x , y ) } , or j ( n ) = min { m ∈ N | ∀ ( a , b ) ∈ Z with | ( b − a ) : ∃ q ∈ {
1, . . . , m } : n ⊥ ( a + q , b + q ) } . Remark 2.1.
In the particular case of an odd difference of the pair elements, there isno pair coprime to an even n because either the first or the second element of the pairwould be even. This trivial case must be excluded. Otherwise j ( n ) would not bedefined for even n .This definition is equivalent to the formulation that the paired Jacobsthal function j ( n ) is the greatest difference m between two pairs in a sequence of consecutive pairswith an even difference of its pair elements, which are coprime to n . j ( n ) = max { m ∈ N | ∃ ( a , b ) ∈ Z with | ( b − a ) : n ⊥ ( a , b ) ∧ n ⊥ ( a + m , b + m ) ∧∀ q ∈ {
1, . . . , m − } : n (cid:54)⊥ ( a + q , b + q ) } .In other words, ( j ( n ) − ) is the greatest length m ∗ = m − (cid:104) a , b (cid:105) m ∗ with an even difference of its pair elements where no pair is coprime to n . Remark 2.2.
The following statements are elementary consequences of the defini-tion 2.1 of the paired Jacobsthal function and describe some interesting properties ofit. Equivalent properties are known for the common Jacobsthal function for integersequences [5, 8]. 3roduct. ∀ n n ∈ N : j ( n · n ) ≥ j ( n ) ∧ j ( n · n ) ≥ j ( n ) .Coprime product. ∀ n n ∈ N > | n ⊥ n j ( n · n ) > j ( n ) ∧ j ( n · n ) > j ( n ) .Greatest common divisor. ∀ n n ∈ N : j ( gcd ( n n )) ≤ j ( n ) ∧ j ( gcd ( n n )) ≤ j ( n ) .Prime power. ∀ n , k ∈ N ∀ p ∈ P : j ( p k · n ) = j ( p · n ) .Prime separation. ∀ n , n ∗ , k ∈ N ∀ p ∈ P | n = p k · n ∗ , p ⊥ n ∗ : j ( n ) = j ( p · n ∗ ) .The last remark implies that the entire paired Jacobsthal function is also determinedby its values for products of distinct primes. The particular case of primorial numbersis therefore most interesting because the function values at these points contain therelevant information for constructing general upper bounds. The paired Jacobsthalfunction of primorial numbers h ( n ) is therefore defined as the smallest natural num-ber m , such that every paired progression (cid:104) a , b (cid:105) m of length m with 2 | ( b − a ) containsat least one pair coprime to the product of the first n primes. Definition 2.2.
Primorial paired Jacobsthal function.
For n ∈ N , the primorial paired Jacobsthal function h ( n ) is defined as h ( n ) = j ( p n ) .In other words, h ( n ) is the smallest length m of a paired progression (cid:104) a , b (cid:105) m with2 | ( b − a ) containing at least one pair coprime to all of the first n primes. h ( n ) = min { m ∈ N | ∀ ( a , b ) ∈ Z with | ( b − a ) : ∃ ( x , y ) ∈ (cid:104) a , b (cid:105) m ∀ i ∈ {
1, . . . , n } : p i ⊥ ( x , y ) } . Goldbach conjecture
Goldbach formulated his original conjecture in a letter to Euler [4] dated June 7, 1742[6, 2]. It has become one of the most famous unsolved mathematical problems. For ourconsiderations, we investigate the best known and sometimes called strong or binaryvariant of the problem.
Conjecture 1.
Goldbach conjecture.
Every even natural number 2 · n with n ∈ N > q , q ∈ P . ∀ n ∈ N > ∃ q , q ∈ P : q + q = · n .4e like to tighten this conjecture slightly and formulate a more constructiveassertion. Corollary 3.1.
Let n ∈ N ≥ , and k n denotes the index of the uniquely determined primep k n ∈ P with p k n + p k n ≤ · n < p k n + + p k n + .The Goldbach conjecture holds if for all n ≥ there exist two primes q , q ∈ P with theconditions p k n < q < p k n and q + q = · n.Proof. This is indeed a tightening because potential prime pairs with q ≤ p k n andtherefore q ≥ p k n are not considered. The smallest example therefor is n = p k n = + = n ≥
6. The examples 4 = + = +
3, 8 = +
5, and 10 = + Proposition 3.2.
Let n ∈ N ≥ , and k n denotes the index of the prime p k n ∈ P withp k n + p k n ≤ · n < p k n + + p k n + .If h ( k ) < p k − p k holds for all k ∈ N ≥ then for all n ≥ , there exist two primesq , q ∈ P with the conditions p k n < q < p k n and q + q = · n. ( ∀ k ∈ N ≥ h ( k ) < p k − p k ) ⇒ ( ∀ n ∈ N ≥ ∃ q , q ∈ P : p k n < q < p k n ∧ q + q = · n ) . Proof.
For 6 ≤ n <
30, we get k n = p k n =
3, and 3 < q <
9. The followingexamples fulfil the requirements: 12 = +
7, 14 = +
7, 16 = +
11, 18 = + = +
13, 22 = +
17, 24 = +
19, 26 = +
19, 28 = + n ≥
30 and the appropriate k n as above. Then k n ≥
3. Let q , q ∈ P be a pair of primes with p k n < q < p k n and q + q = · n . By assumption, we get thefollowing inequations. q > p k n , and q < p k n < p k n + . q = · n − q ≥ p k n + p k n − q > p k n + p k n − p k n = p k n , and q = · n − q < p k n + + p k n + − q ≤ p k n + + p k n + − p k n + = p k n + because q is prime and therefore q ≥ p k n + > p k n .For the primality of q and q with p k n < q , q ∈ N < p k n + and q + q = · n ,it is necessary and sufficient that q ⊥ p i and q = · n − q ⊥ p i for all i =
1, . . . , k n .Furthermore, q ⊥ p i if and only if − q = q − · n ⊥ p i .We now consider the paired progression (cid:104) p k n , p k n − · n (cid:105) p kn − p kn − with an obvi-ously even difference of its corresponding pair elements. According to the assumption h ( k ) < p k − p k for all k ∈ N ≥
3, we get h ( k n ) ≤ p k n − p k n −
1, and every pairedprogression of length p k n − p k n − p i , i =
1, . . . , k n . Andso it does for (cid:104) p k n , p k n − · n (cid:105) p kn − p kn − and at least one of its pairs fulfills the require-ments of the proposition. 5 win prime conjecture The twin prime conjecture is another best-known conjecture in number theory. It as-serts that there are infinitely many prime twins, i.e. two primes with the difference 2.The true origin of this assertion is unrecorded. Euclid proved the infinitude of primes.Sometimes, the twin prime conjecture was ascribed to Euclid therefore but with no ref-erence. It has become common to regard the relation to the conjecture of de Polyniac[1] and consider this as the origin of the twin prime conjecture, too.The conjecture of de Polyniac [2, 7] asserts that every even natural number can bewritten in infinitely many ways as the difference of two consecutive primes. The twinprime conjecture is indeed included in this assertion when the even number is chosento be 2.
Conjecture 2.
Twin prime conjecture.
There exist infinitely many pairs of primes q , q ∈ P with the difference 2. |{ q , q ∈ P | q − q = }| = ∞ . Prime pairs conjecture
The question of the infinitude of prime pairs with a fixed even difference is also ageneralisation of the twin prime conjecture and nevertheless a weaker form ofde Polyniac’s conjecture [6, 2, 7]. We investigate this general conjecture because itis closely related to the paired Jacobsthal function in a similar way as we proved forthe Goldbach conjecture.
Conjecture 3.
Prime pairs conjecture.
For every even natural number d = · n with n ∈ N , there exist infinitely many pairsof primes q , q ∈ P the difference of which is d . ∀ n ∈ N : |{ q , q ∈ P | q − q = · n }| = ∞ . Corollary 3.3.
If the prime pairs conjecture holds, so does the twin prime conjecture.Proof.
The twin prime conjecture represents the specific case n = Remark 3.1.
For n =
2, every prime pair q , q ∈ P with q > q − q = · n mustbe a pair of consecutive primes because ( q + q ) /2 is divisible by 3. The conjecture ofde Polyniac and the prime pairs conjecture are even equivalent for n ≤
2, therefore.6e tighten the prime pairs conjecture in a similar, constructive way as we did forGoldbach’s conjecture.
Corollary 3.4.
The prime pairs conjecture holds if for every natural number n ∈ N and everyprime p ∈ P with p > · n, there exists a pair of primes q , q ∈ P with the conditionsp < q < p and q − q = · n.Proof. Given n , we choose an arbitrary p ∈ P > · n . By assumption, there exists a firstpair q , q ∈ P with q − q = · n and q > p > · n .Given any prime pair q i , q j ∈ P with q j − q i = · n and q i > · n , we choose another p ∈ P > q i > · n . Then there exists a pair q i + , q j + ∈ P with q j + − q i + = · n and q i + > p > q i .Infinitude follows by induction.If the same upper bound of the primorial paired Jacobsthal function, which wasrelated to the tightened Goldbach conjecture in proposition 3.2, holds then the abovetightened prime pairs conjecture holds as well. Proposition 3.5.
If h ( k ) < p k − p k holds for all k ∈ N ≥ then for every natural numbern ∈ N and every prime p ∈ P with p > · n, there exists a pair of primes q , q ∈ P with theconditions p < q < p and q − q = · n.Proof. p = p = p k > · n , n ∈ N with k <
3. Then n = q = q = n ∈ N , we require k ≥ p k ∈ P > · n . Let q , q ∈ P be a pair of primes with p k < q < p k and q − q = · n . By assumption, weget the following inequations. q > p k , and q < p k < p k + . q = · n + q > q > p k , and q = · n + q < p k + q < p k + p k < + · p k + p k = ( + p k ) < p k + .For the primality of q and q with p k < q , q ∈ N < p k + and q − q = · n , it isnecessary and sufficient that q ⊥ p i and q = · n + q ⊥ p i for all i =
1, . . . , k .We now consider the paired progression (cid:104) p k , p k + · n (cid:105) p k − p k − with an obviouslyeven difference of its corresponding pair elements. According to the assumption, weget h ( k ) ≤ p k − p k −
1, and every paired progression of length p k − p k − p i , i =
1, . . . , k . And so it does for (cid:104) p k , p k + · n (cid:105) p k − p k − and at leastone of its pairs fulfills the requirements of the proposition.7 Conclusions
In the previous section, we alleged three new conjectures. Two of them are tighteningsof assumptions well-known for a long time. These are the Goldbach conjecture and theinfinitude of prime pairs with a fixed even difference. The third case concerns an upperbound of the primorial paired Jacobsthal function which we defined beforehand.Below, we provide explicit formulations of these new conjectures.
Conjecture 4.
Tightened Goldbach conjecture.
Let n ∈ N ≥
6, and k n denotes the index of the uniquely determined prime p k n ∈ P with p k n + p k n ≤ · n < p k n + + p k n + . Then for all n ≥
6, there exist two primes q , q ∈ P with the conditions p k n < q < p k n and q + q = · n . ∀ n ∈ N ≥ p k n + p k n ≤ · n < p k n + + p k n + ∃ q , q ∈ P : p k n < q < p k n ∧ q + q = · n .This conjecture was verified for all n with 12 ≤ · n ≤ . In the ancillary file"goldbach.lis", we exemplarily provide an exhaustive list of all of the always smallestcorresponding pairs ( q , q ) with q + q = · n for 12 ≤ · n ≤ Conjecture 5.
Tightened prime pairs conjecture.
For every natural number n ∈ N and every prime p ∈ P with p > · n , there existsa pair of primes q , q ∈ P with the conditions p < q < p and q − q = · n . ∀ n ∈ N ∀ p ∈ P , p > · n ∃ q , q ∈ P : p < q < p ∧ q − q = · n .This conjecture was verified for all n with 2 · n ≤ and all 3 ≤ p < . Inthe ancillary file "pairs.lis", we exemplarily provide an exhaustive list of all of thealways smallest corresponding pairs ( q , q ) with q − q = · n for 2 · n ≤
100 andall 3 ≤ p < Conjecture 6.
Upper bound of the primorial paired Jacobsthal function.
Let n ∈ N ≥
3. Then h ( n ) < p n − p n .8he following scheme resumes and depicts the results of the previous sections.New conjectures are highlighted in grey. Upper boundof the primorialpaired Jacobsthalfunction h ( n ) < p n2 - p n , n ‡ Conjecture 6 (cid:222) P r o p o s i t i o n . TightenedGoldbachconjecture
Conjecture 4 (cid:222)
Corollary 3.1
Goldbachconjecture
Conjecture 1 (cid:222) P r opo s i t i on . Tightenedprime pairsconjecture
Conjecture 5 (cid:222)
Corollary 3.4
Infinitudeof prime pairesfor every evendifference
Conjecture 3 (cid:222)
Corollary 3.3
Infinitudeof prime twins
Conjecture 2
Figure 1: Scheme of proved inferences.The main result of this paper is summarised in the following concluding theorem.
Theorem 4.1.
The conjectured upper bound of the primorial paired Jacobsthal function issufficient for the truth of the Goldbach conjecture and of the infinitude of prime pairs for everyeven difference.Proof.
This theorem follows from the propositions 3.2 and 3.5 and the corollaries 3.1and 3.4.
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