THE BINARY GOLDBACH CONJECTURE IS ALSO TRUE

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THE BINARY GOLDBACH CONJECTURE IS ALSO TRUE

Ricardo Barca

To cite this version:

Ricardo Barca. THE BINARY GOLDBACH CONJECTURE IS ALSO TRUE. 2020. �hal-02924032�

THE BINARY GOLDBACH CONJECTURE IS ALSO TRUE

Ricardo G. Barca

Abstract

The binary Goldbach conjecture asserts that every even integer greater than 4 is the sum of two primes. In a preceding paper we have proved that there exists a positive integer K

such that every even integer x > p

can be expressed as the sum of two primes, where p

is the kth prime number and k > K

. In this paper we provide an estimation for K

, and from this result it follows that the binary Goldbach conjecture is true.

1 Introduction

This paper is the continuation of the author’s work [2] and proves the binary Goldbach conjecture.

Main Theorem . Every even integer x greater than 4 can be expressed as the sum of two primes.

2 The behaviour of the h-density within the Right block of the partition of the first period of S k , for sufficiently large k

Let S k (k ≥ 4) be a given partial sum of the series P

s k . For every partial sum S h from level h = 1 to level h = k, let us consider the interval I[1, m k ] h , the Left interval I[1, p ² _k ] h and the Right interval I[p ² _k + 1, m k ] h . Recall the notation c ^L _h

to denote the number of permitted h-tuples within the Left interval I[1, p ² _k ] _h , and the notation c ^R _h

to denote the number of permitted h-tuples within the Right interval I[p ² _k + 1, m _k ] _h . Recall the notation m _h for the period of the partial sum S _h and c _h for the number of permitted h-tuples within a period of the partial sum S _h (1 ≤ h ≤ k).

Furthermore, recall the notation c

_h to denote the number of permitted h-tuples within the interval I[1, m _k ] _h of every partial sum S h (1 ≤ h ≤ k), and recall the notation δ h , δ _h ^L

and δ _h ^R

for the density of permitted h-tuples within the intervals I[1, m k ] h , I[1, p ² _k ] h and I[p ² _k + 1, m k ] h , respectively.

The following lemma shows that for every partial sum S h (1 ≤ h ≤ k), the h-density within the Left interval I[1, p ² _k ] h cannot be equal to the h-density within the Right interval I[p ² _k + 1, m k ] h .

Lemma 2.1. Let S k (k ≥ 4) be a given partial sum of the series P

s k . Let us consider the interval I[1, m k ] h (whose size is the period m _k of S _k ), the Left interval I[1, p ² _k ] _h and the Right interval I[p ² _k + 1, m _k ] _h , in every partial sum S _h from level h = 1 to level h = k.

For every partial sum S _h , we have δ _h ^L

6= δ ^R _h

.

Proof. Step 1. By [2, Remark 6.1], the positions of consecutive permitted 1-tuples in the partial sum S ₁ differ by 2.

It follows that the number of permitted h-tuples in every Left interval I[1, p ² _k ] h (1 ≤ h ≤ k) is less than the size of the interval; that is, c ^L _h

< p ² _k (1 ≤ h ≤ k).

Step 2. Let us consider a given partial sum S h , where 1 ≤ h < k. Consider the number of permitted h-tuples in the Left interval I[1, p ² _k ] h , denoted by c ^L _h

, and the number of permitted h-tuples in the Right interval I[p ² _k + 1, m k ] h , denoted by c ^R _h

. By [2, Lemma 6.1], we have c ^L _h

+c ^R _h

= c _h p _h+1 p _h+2 · · · p _k ; therefore, if c ^L _h

is a multiple of p _k , then c ^R _h

is also a multiple of p _k . In this case, (c ^L _h

/p _k ) is a whole number, so (c ^L _h

/p _k )/p _k is a reduced fraction since c ^L _h

< p ² _k , by Step 1. Hence, it is easy to check that this fraction is not equal to (c ^R _h

/p _k )/(m _k−1 − p _k ) since (c ^R _h

/p k ) is also a whole number, and m _k−1 − p k is not a multiple of p k .

On the other hand, if c ^L _h

is not a multiple of p k , then c ^L _h

/p ² _k is a reduced fraction, and it is easy to check that this fraction is not equal to c ^R _h

/(m k −p ² _k ) since m k − p ² _k is not a multiple of p ² _k . In either case, we proved that c ^L _h

/p ² _k is not equal to c ^R _h

/(m k − p ² _k ).

Step 3. Now, let us consider the partial sum S k . Consider the number of permitted k-tuples in the Left interval

I[1, p ² _k ], denoted by c ^L _k

, and the number of permitted k-tuples in the interval I[1, m k ] (complete period of

S _k ), denoted by c _k . By [2, Proposition 2.3], we have c _k = (p ₁ − 1)(p ₂ − 2)(p ₃ − 2) · · · (p _k − 2). Now, if c ^L _k

is

not a multiple of p _k , we can see that c ^L _k

/p ² _k is a reduced fraction, and it is easy to check that this fraction

cannot be equal to c _k /m _k since m _k is a squarefree integer.

On the other hand, if c ^L _k

is a multiple of p k , then (c ^L _k

/p k ) is a whole number, and (c ^L _k

/p k )/p k is a reduced fraction since c ^L _k

< p ² _k , by Step 1. From [2, Proposition 2.1] and [2, Proposition 2.3], it follows that

c k

m k

= (p 1 − 1) (p 2 − 2) (p 3 − 2) · · · (p k−1 − 2) (p k − 2) p 1 p 2 p 3 · · · p _k−1 p k

=

=

p 1 − 1 p ₁

p 2 − 2 p ₂

p 3 − 2 p ₃

· · ·

p _k−1 − 2 p _k−1

p k − 2 p _k

,

and by shifting the denominators to the right, we obtain

c k

m _k =

p 2 − 2 p ₁

p 3 − 2 p ₂

· · ·

p _k−1 − 2 p _k−2

p k − 2 p _k−1

1 p _k .

Note that p 1 cannot be canceled with any numerator of the fractions in parentheses, which are all odd integers.

Thus, it is easy to check that this fraction is not equal to the reduced fraction (c ^L _k

/p k )/p k .

In either case, the proportion of permitted k-tuples in the interval I[1, p ² _k ], given by c ^L _k

/p ² _k , is not equal to the proportion of permitted k-tuples in the interval I[1, m k ], given by c k /m k . Thus, if c ^L _k

/p ² _k > c k /m k , it must be true that c ^R _k

/(m k − p ² _k ) < c k /m k , and vice versa. This result implies c ^L _k

/p ² _k 6= c ^R _k

/(m k − p ² _k ).

Step 4. We prove the lemma. From Steps 2 and 3, for every partial sum S _h (1 ≤ h ≤ k), it follows that c ^L _h

/p ² _k 6=

c ^R _h

/(m _k − p ² _k ). Multiplying by p _h , we obtain p _h c ^L _h

/p ² _k 6= p _h c ^R _h

/(m _k − p ² _k ); thus, δ ^L _h

6= δ _h ^R

.

Now, with regard to the Right block of the partition, the following question arises: For sufficiently large k, could it be that δ ^R _h

< δ h (δ ^R _h

> δ h ) for every partial sum S h from h = 1 to h = k? We answer this question in the following paragraphs.

Given the partial sums S h from h = 1 to h = k, since p ² _k = o(m k ), the size of every Right interval I[p ² _k + 1, m k ] h

approximates the size of the interval I[1, m k ] h more and more closely as k → ∞. Therefore, we can take k sufficiently large that the size of the Left interval I[1, p ² _k ] is negligible compared to the size of the interval I[1, m k ], thus, almost all the permitted h-tuples in every interval I[1, m k ] h (1 ≤ h ≤ k) belong to the Right interval I[p ² _k + 1, m k ] h . It follows that, for every level from h = 1 to h = k, the proportion of permitted h-tuples in the Right interval I[p ² _k + 1, m k ] h , given by c ^R _h

/(m _k − p ² _k ), will be very close to the proportion of permitted h-tuples in the interval I[1, m _k ] _h , given by c

_h /m _k , regardless of the combination of selected remainders in the sequences s _h that form the partial sum S _k ; that is, δ _h ^R

will be very close to δ h .

Now, let S k be a given partial sum of the series P

s k , where k is sufficiently large that the size of the Left interval I[1, p ² _k ] is negligible compared to the size of the interval I[1, m k ]. Consider S h , where 1 ≤ h < k, and recall the notation r, r

for the selected remainders modulo p h+1 within the period of the sequence s h+1 . Furthermore, recall the notation η r , η r

for the number of permitted h-tuples within the Right interval I[p ² _k + 1, m k ] h whose indices belong to the residue classes [r], [r

] modulo p h+1 , respectively.

When we juxtapose the elements of the sequence s _h+1 to each h-tuple of S _h , by [2, Proposition 2.2], every permitted h-tuple of S _h whose index is congruent to a selected remainder of s _h+1 modulo p _h+1 is removed. Thus, the indices of the permitted h-tuples removed from the interval I[1, m _k ] _h belong to one of the residue classes [r] or [r

] modulo p _h+1 . Since the permitted h-tuples within the interval I[1, m _k ] _h of S _h are distributed uniformly over the residue classes modulo p _h+1 , by [2, Proposition 2.4] and [2, Corollary 2.5], it follows that the fraction of permitted h-tuples that are removed from the interval I[1, m k ] h is equal to 2/p h+1 .

On the other hand, the fraction of permitted h-tuples that are removed from the Right interval I[p ² _k + 1, m k ] h of S h

is equal to (η r + η r

)/c ^R _h

. However, the distribution of the permitted h-tuples within the Right interval I[p ² _k + 1, m k ] h

of S h over the residue classes modulo p h+1 , clearly, is not uniform, since the size of I[p ² _k + 1, m k ] h is not a multiple of m _h . Thus, depending on the selected remainders r, r

modulo p _h+1 , it could be either (η _r + η _r

)/c ^R _h

> 2/p _h+1 or (η r + η r

)/c ^R _h

< 2/p h+1 .

We can see that for k sufficiently large that the size of the Left interval I[1, p ² _k ] is negligible compared to the size

of the interval I[1, m k ], for every level from h = 1 to h = k, the density δ _h ^R

will be very close to the average δ h ,

and furthermore, for every level transition p h → p h+1 (1 ≤ h < k), the fraction of permitted h-tuples within the

Right interval interval I[p ² _k + 1, m _k ] _h of S _h that are transferred to the Right interval interval I[p ² _k + 1, m _k ] _h+1 of S _h+1

as permitted (h + 1)-tuples could be above or below the average fraction (p _h+1 − 2)/p _h+1 given by [2, Lemma 6.2])

(depending on the selected remainders r, r

modulo p _h+1 in s _h+1 ). Therefore, the answer to the question above is ‘no’,

and we conclude the following.

Remark 2.1. With regard to the Right block of the partition of the first period of S k , for k sufficiently large that the size of the Left interval I[1, p ² _k ] is negligible compared to the size of the interval I[1, m k ], between h = 1 and h = k there must be levels for which δ _h ^R

> δ h mixed with levels for which δ _h ^R

< δ h , regardless of the combination of selected remainders in the sequences s h that form the partial sum S k .

Remark 2.2. For a level k sufficiently large that the size of the Left interval I[1, p ² _k ] is negligible compared to the size of the interval I[1, m k ], from the preceding remark, Lemma 2.1 and [2, (21)], in the Left block of the partition of the first period of S k there must be levels for which δ _h ^L

< δ h mixed with levels for which δ ^L _h

> δ h , between h = 1 and h = k.

3 Estimating a value for K _α

For every partial sum S h (1 ≤ h ≤ k) recall the notation {δ _h ^L

} to denote the set of values of δ _h ^L

, for all the combinations of selected remainders in the sequences that form the partial sum S k .

Given the level transition p h → p h+1 (1 ≤ h < k), recall the notation F _h,h+1 ^L

for the fraction of the permitted h-tuples within the Left interval I[1, p ² _k ] h of S h that are transferred to the Left interval I[1, p ² _k ] h+1 of S h+1 as permitted (h + 1)-tuples. Recall that by [2, Lemma 6.2], the average of F _h,h+1 ^L

is equal to (p h+1 − 2)/p h+1 .

Remark 3.1. Let us consider a given level h = h

(1 < h

< k). Assume that for h ≥ h

, if p h → p h+1 is a level transition of order greater than 2, in each period (of size p h+1 ) of every sequence s h+1 there are 3 selected remainders (the same in every period of the sequence), and if p _h → p _h+1 is a level transition of order 2, in each period (of size p _h+1 ) of every sequence s _h+1 there are 2 selected remainders (the same in every period of the sequence). Under the preceding assumption, if p _h → p _h+1 is a level transition of order greater than 2, the fraction of permitted h-tuples within the interval I[1, m _k ] _h of S _h that are transferred to the interval I[1, m _k ] _h+1 of S _h+1 as permitted (h + 1)-tuples is equal to (p _h+1 − 3)/p _h+1 , and if p _h → p _h+1 is a level transition of order 2, the fraction of permitted h-tuples within the interval I[1, m k ] h of S h that are transferred to the interval I[1, m k ] h+1 of S h+1 as permitted (h+ 1)-tuples is equal to (p h+1 − 2)/p h+1 , where h ≥ h

, regardless of the combination of selected remainders in S k (see the proof of [2, Lemma 6.2]). Thus, in every level transition p h → p h+1 (h

≤ h < k) of order greater than 2, the increase of δ h in this case would be given by

δ h+1 = δ h

p h+1 − 3 p h+1

p h+1

p h

= δ h

p h+1 − 3 p h

, (1)

(see [2, Remark 6.4]).

Now, under the condition assumed in Remark 3.1, the number of permitted h-tuples removed from I[1, m k ] h

between h = h

and h = k is greater than in the case where there are only two selected remainders in the sequences s _h (h

< h ≤ k), for sufficiently large k, so, the density of permitted h-tuples within the interval I[1, m _k ] _h increases more slowly as h goes from level h

to level k. We see that, under this condition, the density of permitted k-tuples within the interval I[1, m _k ] _k of S _k , using (1), would be

δ h









k−1

Y

h=h

p

=2

p h+1 − 2 p _h

















k−1

Y

h=h

p

>2

p h+1 − 3 p _h









= δ h

k−1

Y

h=h

p

>2

p h+1 − 3 p _h

. (2)

With regard to the Left intervals I[1, p ² _k ] h (h

≤ h < k), since in our case the condition assumed in Remark 3.1 is not satisfied (see [2, Definition 2.7]), it follows that for every level transition p h → p h+1 of order greater than 2 between h = h

and h = k, the fraction F _h,h+1 ^L

must be much closer to the average fraction (p h+1 − 2)/p h+1 (given by [2, Lemma 6.2]) than to (p h+1 − 3)/p h+1 (the corresponding average fraction which was obtained under the condition assumed in Remark 3.1). Then, if the number and proportion of level transitions p _h → p _h+1 of order greater than 2 between h = h

and h = k is sufficiently large, the behaviour of δ ^L _h

between h = h

and h = k will be more similar to the current behaviour of δ _h in this paper (given by [2, Lemma 3.2]) than to the behaviour given by (1) under the condition assumed in Remark 3.1, regardless of the combination of selected remainders in S _k . Thus,

δ k = δ h

k−1

Y

h=h

p h+1 − 2 p _h

> min{δ ^L _k

} > δ h

k−1

Y

h=h

p

>2

p h+1 − 3 p _h

(3)

for sufficiently large k.

Now, by [2, Lemma 7.1], there exists K α such that δ _k ^L

> δ 4 if k > K α , for every combination of selected remainders

in the sequences s h that form the partial sum S k . Suppose that we take K α = 3246 and let h

= 5. Since there are

2775 or more level transitions of order greater than 2 between h = 5 and h = k (k > K α ), it seems reasonable to assume that the bounds in (3) are satisfied for every k > K α .

However, suppose that someone ask the following question: for a level k > K α , could

min{δ ^L _k

} ≤ δ ₅

k−1

Y

h=5 p

>2

p _h+1 − 3 p h

, (4)

for an specific combination of selected remainders in S _k ? The following arguments show that the answer to the preceding question is negative.

On the one hand, the average h-density within the Left interval I[1, p ² _k ] _h increases slowly as h goes from level h

to level k (see [2, Lemma 3.2]). Furthermore, for a partial sum S _k where k > K _α = 3246, clearly, the size of the Left interval I[1, p ² _k ] is negligible compared to the size of I[1, m k ], so, by Remark 2.2, as we go from level h

to level k, we must find levels h where δ ^L _h

< δ h mixed with levels where δ _h ^L

> δ h (see [2, Remark 7.2]).

On the other hand, it is easy to check that

δ 5 k−1

Y

h=5 p

>2

p h+1 − 3 p _h

= o δ 5 k−1

Y

h=5

p h+1 − 2 p _h

!

as k → ∞,

that is

δ ₅

k−1

Y

h=5 p

>2

p _h+1 − 3 p h

= o (δ _k ) . (5)

Thus, for k > K _α = 3246, we can check that the left-hand side of (5) is less than half of δ _k . Then, if (4) is satisfied, δ _h ^L

will be less than half of δ _k for all the levels between h = 1 and h = k, where k > K _α = 3246, contradicting Remark 2.2. Therefore, the answer to the preceding question is ‘no’, and we conclude that the estimation (3) holds for every k > K α = 3246, regardless of the combination of selected remainders in the sequences s h that form the partial sum S k .

4 Conclusion

Let = 2.5 × 10

. For level k > K α = 3246 we have p ² _k > 900660121, and consequently, for the Left interval I[1, p ² _k ] 5 , using [2, Lemma 5.2], it is easy to check that

δ 5 − < δ ₅ ^L

< δ 5 + ,

for every combination of selected remainders in S _k . On the other hand, we can check that

δ 5 k−1

Y

p

h=5

>2

p _h+1 − 3 p h

> δ 5 + (k > K α ). (6)

Hence, by (3) and (6) we obtain

δ k = δ 5 k−1

Y

h=5

p h+1 − 2 p h

> min{δ ^L _k

} > δ 5 k−1

Y

p

h=5

>2

p h+1 − 3 p h

> δ 5 + (k > K α ). (7)

Then, in view of (7) and [2, (23)], for every combination of selected remainders in S k , where k > K α , we can write

δ _k ^L

= δ 5 1 + θ k k−1

Y

h=5

p h+1 − 2 p _h

− 1

!!

> δ 5 > δ 4 , (8)

where θ k > 0, θ k 6= 1 is some real number which depends on the combination of selected remainders in S k . Thus,

by (8), we have obtained

Lemma 4.1. Given the partial sum S k , where k > K α = 3246, for every combination of selected remainders in S k

we have δ ^L _k

> δ 4 .

Proof of the Main Theorem. By [2, Lemma 7.1], there exists K α such that δ _k ^L

> δ 4 if k > K α , for every combi- nation of selected remainders in the sequences s h that form the partial sum S k . Furthermore, by Lemma 4.1, taking K α = 3246, the preceding statement is satisfied. Therefore, by [2, Lemma 8.3], every even integer x > p ² _k is the sum of two primes, where k > K α = 3246. That is, if x is greater than p ² ₃₂₄₆ = 900660121 it can be expressed as the sum of two primes.

Now, it is a known fact that the strong Goldbach conjecture has already been verified for all even numbers x ≤ 4 × 10 ¹⁷ [1]. Therefore, we conclude that every even number x > 4 can be expressed as the sum of two primes;

thus, the binary Goldbach conjecture is proved.

Acknowledgements

The author wants to thank Dra. Patricia Quattrini (Departamento de Matematicas, FCEyN, Universidad de Buenos Aires) for helpful conversations and Dr. Hendrik W. Lenstra (Universiteit Leiden, The Netherlands) for his extremely useful suggestions.

References

[1] T. Oliveira e Silva, Goldbach conjecture verification, http://www.ieeta.pt/~tos/goldbach.html .

[2] Ricardo Barca, Every sufficiently large even number is the sum of two primes, https://hal.archives-ouvertes.fr/

hal-02075531.

Ricardo G. Barca

Universidad Tecnologica Nacional Buenos Aires (Argentina)

E-mail address: rbarca@frba.utn.edu.ar