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Journals / Ikonion Journal of Mathematics / 2026 / Volume: 8 Issue: 1

Empirical Verification of the Weak Goldbach Conjecture for Odd Integers up to 4300 Digits(≈ 5.075×〖10〗^4299 ): A Novel Computational Approach

Daniel Sankei(Meru University of Science and Technology)0009-0008-8237-8563
Loyford Njagi(Meru University of Science and Technology)0009-0000-1545-2569
Josephine Mutembei(Meru University of Science and Technology)0009-0007-6154-625X
Grace Gakii(MERU UNIVERSITY OF SCIENCE AND TECHNOLOGY)0000-0002-3108-6471
Pages
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Publication Date
2026-07-16
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

This work highlights the potential of computational methods in number theory.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The Weak Goldbach Conjecture asserts that every odd integer greater than 5 can be expressed as the sum of three primes.
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Key Findings & Data:While Helfgott (2013) provided a theoretical proof, computational verification remains critical for extremely large integers.
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Conclusion & Impact:This work highlights the potential of computational methods in number theory.
🔬Design:Research Article
📍Scope:2013 Verileri
⚙️Variables:Weak Goldbach Conjecture · Prime Partitioning · Computational Number Theory

Abstract

The Weak Goldbach Conjecture asserts that every odd integer greater than 5 can be expressed as the sum of three primes. While Helfgott (2013) provided a theoretical proof, computational verification remains critical for extremely large integers. This study presents a novel computational framework that extends empirical validation to odd integers with up to 4300 digits (≈5.075×〖10〗^4299), a significant leap beyond previous limits. Our approach integrates a probabilistic primality test with an optimized partitioning algorithm, systematically exploring configurations of the form (e_1+p,e_2+p,c), where e_1,e_2 are even, p is odd, and c=O-(e_1+e_2+2p), ensuring all terms are prime. Utilizing optimized high-precision arithmetic and parallel processing techniques, we verified the conjecture for random odd integers O with 9≤d≤4300 digits, achieving runtimes under 43 minutes for the largest cases. Independent verification confirmed the primality of all triplet components, ensuring mathematical rigor. This work highlights the potential of computational methods in number theory.

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Cite This Article

Sankei, D., Njagi, L., Mutembei, J., Gakii, G. (2026). Empirical Verification of the Weak Goldbach Conjecture for Odd Integers up to 4300 Digits(≈ 5.075×〖10〗^4299 ): A Novel Computational Approach. Ikonion Journal of Mathematics, Volume: 8 Issue: 1. https://doi.org/10.54286/ikjm.1659107

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References (12)

  1. Al-Ameen, T., & Muhi, I. (2022, March). A rudimentary proof on Goldbach conjectures. In 2022 IEEE 2nd International Conference on Information Communication and Software Engineering (ICICSE) (pp. 13-17). IEEE.
  2. Helfgott, H. A. (2013). The ternary Goldbach conjecture is true. arXiv preprint arXiv:1312.7748.
  3. Saouter, Y. (1998). Checking the odd Goldbach conjecture up to 10²⁰. Mathematics of computation, 67(222), 863-866.
  4. Oliveira e Silva, T., Herzog, S., & Pardi, S. (2014). Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4×〖10〗^18. Mathematics of Computation, 83(288), 2033-2060.
  5. Rabin, M. O. (1980). Probabilistic algorithm for testing primality. Journal of Number Theory, 12(1), 128138.
  6. Miller, G. L. (1976). Riemann's hypothesis and tests for primality. Journal of Computer and System Sciences, 13(3), 300-317.
  7. https://colab.research.google.com/
  8. Hardy, G. H., & Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press.
  9. Richstein, J. (2001). Verifying the Goldbach conjecture up to 4⋅ 10¹⁴. Mathematics of computation, 70(236), 1745-1749.
  10. Van Rossum, G., & Drake, F. L. (2009). Python 3 Reference Manual. CreateSpace.
  11. Granlund, T., et al. (2015). The GNU Multiple Precision Arithmetic Library.
  12. Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., & Zimmermann, P. (2007). MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Transactions on Mathematical Software, 33(2), 13.