Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

A Solution to Goldbach’S Conjecture

Abhijit Manohar

Kolhapur, Maharashtra, India

Abstract: The discussion revisits Goldbach’s conjecture by outlining a numerical approach that expresses every even integer as the sum of two prime numbers through relationships based on the form 6k plus or minus 1. The analysis shows how pairs of constants can be used to generate corresponding primes for any even value by drawing on the structure of integers within modular classes. Examples from small values and extended ranges, including even numbers between 1000 and 1100, demonstrate how specific combinations of k and k' yield valid prime pairs. The argument is supported by broader computational work that has explored the conjecture up to very large magnitudes, reinforcing the view that even integers consistently align with prime pair representations within this framework.

Keywords: Goldbach conjecture, prime numbers, LeonhardEuler, even integers, tested for 4 x 10¹⁸

Received November 08, 2025; Revised December 10, 2025; Accepted December 17, 2025

Citation: Abhijit Manohar, A Solution to Goldbach’S Conjecture. American Journal of Mathematical Analysis, 11 (1). 12-13. January 2025. doi:10.12691/ajma-11-1-1

1. Introduction

Goldbach proposed a conjecture in the margin of his letter to Leonhard Euler, that every integer greater than 2 can be written as the sum of three primes. Later the conjecture was changed to – Every positive even integer is the sum of 2 prime numbers. Since µ, ε → 1 to ∞ for n →1 to ∞ the proof shows that the formula works for all even integers.

2. Proof

Prime numbers greater than 3 are given by (6k +/-1);

2n - (6k +/-1) = (6k' +/-1); where k, k’ are 2 constants

n →1 to ∞

2n = (6k +/-1) |k = µ + (6k' +/-1) |k=ε ≥ 10 where µ, ε → 1 to ∞

2n = 6(k +k') +/-2 V 6(k+k')

For even integers < 10, 8= 5 + 3, 6 = 3 + 3, 4 = 2 + 2.

Example for even number 26, k’ = 1, k = 3 where k and k’ are 2 constants is shown in Table 1 above. The primes are 19 and 7. This was found from the equation 2n = 6(k + k’) + 2. 24 is found from the equation 2n = 6(k + k’) and 22 is found from the equation 2n = 6(k + k’) - 2. This covers 3 consecutive even integers for (k + k’) = 4. But this is true for all values of (k + k’) ≥ 2. (k + k’) → 2 to ∞. Hence the conjecture that every even integer is sum of 2 prime numbers is true for all even integers.

Table 1.

Table 2 below shows that the proof works for even integers from 1000 to 1100. For even number 1046, we use 2n = 6(k + k’) + 2. K = 173 and k’ = 1. The primes are 1039 and 7.

Table 2.

T. Oliveira e Silva ran a verification for Goldbach conjecture up to 4 x 10¹⁸ and found that 3,325,581,707,333,960,528 is the smallest number which cannot be written as the sum of 2 primes where one is smaller than 9781 [3].

3. Conclusion

Since µ, ε → 1 to ∞ for n →1 to ∞ the proof shows that the formula works for all even integers.

References

[1]   | In the printed version published by P. H. Fuss [1] 2 is misprinted as 1 in the marginal conjecture.
[2]   | "Letter XLIV, Euler to Goldbach" (PDF). Correspondence of Leonhard Euler. Mathematical Association of America. 30 June 1742. Archived (PDF) from the original on 2024-09-17. Retrieved 2025-01-19.

[3]  
Oliviera e Silva, Tomas; Herzog, Siegfried; Pardi, Silvio (July 2014). "Empirical Verification of the Even Goldbach Conjecture and Computation of up to 4 · 1018" (PDF). Mathematics of Computation. 83 (288). American Mathematical Society: 2033–2068. Archived (PDF) from the original on 15 June 2025. Retrieved 16 July 2025.