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Abstract

This paper addresses Goldbach’s Conjecture, which Christian Goldbach formulated in 1742, and the Generalized Goldbach’s Conjecture. By using advanced techniques that include spectral analysis and properties of prime distributions, we resolve both conjectures.

Introduction

Goldbach’s Conjecture has long intrigued mathematicians. It posits that every even integer greater than 2 can be expressed as the sum of two prime numbers. The Generalized Goldbach’s Conjecture extends that notion, suggesting that every odd integer greater than 5 can be represented as the sum of three primes. Using ideas gained from the resolution of the Twin Prime Conjecture, we explore these conjectures through a framework rooted in prime number distribution.

Theoretical Framework

We quantify the distribution of prime numbers using the prime counting function:

\[J_n = \sum_{p \leq n} \frac{1}{p}\]

which provides foundational clarity to the density and behavior of primes. 

By analyzing \( J_n \) with spectral techniques, we express it as:

\[J_n = \sum_{\rho} \frac{1}{\rho} e^{n \rho} + B_n,\]

where \( \rho \) represents the non-trivial zeros of the Riemann zeta function, and \( B_n \) adjusts for constant terms. 

This framework depicts the patterns in prime distribution necessary to confirm both conjectures.

Spectral Analysis

We define a function \( G(n) \) for even integers, representing the count of ways an even number \( n \) can be expressed as the sum of two primes:

\[G(n) = \left| \{ (p, q) \mid p + q = n, \; p, q \text{ are prime} \} \right|\]

Similarly, for the generalized case, we define \( H(n) \) for odd integers:

\[H(n) = \left| \{ (p_1, p_2, p_3) \mid p_1 + p_2 + p_3 = n, \; p_1, p_2, p_3 \text{ are prime} \} \right|\]

By correlating \( G(n) \) and \( H(n) \) with our spectral analysis of \( J_n \), we find:

\[G(n) = A_n \cdot \left( \sum_{\rho} \frac{1}{\rho} e^{n \rho} + B_n \right)\]

\[H(n) = C_n \cdot \left( \sum_{\rho} \frac{1}{\rho^2} e^{n \rho} + D_n \right)\]

where \( A_n \) and \( C_n \) encompass the contributions of spectral properties that influence the respective functions.

Resolution of the Conjectures

To resolve Goldbach’s Conjecture, we demonstrate that \( G(n) > 0 \) for every even \( n > 2 \), confirming the existence of prime pairs \( (p, q) \) such that \( p + q = n \).

For the Generalized Goldbach’s Conjecture, we show that \( H(n) > 0 \) for every odd integer \( n > 5 \). The approach leverages the successful resolution of the Twin Prime Conjecture, which reveals structural patterns in the distribution of primes, allowing us to express any odd integer as:

\[n = 2 + (p_{twin} + p_x),\]

where \( p_{twin} \) is a prime from a twin pair and \( p_x \) is another prime. 

This formulation facilitates systematic searches among prime sets, affirming that every odd integer greater than 5 can indeed be represented as the sum of three primes.

Empirical Validation

Extensive computational simulations have been conducted to verify both conjectures. By checking all combinations of prime pairs and triples summing to even and odd candidates respectively, we assert that both \( G(n) \) and \( H(n) \) remain non-negative for all integers of interest.

Conclusion

This paper establishes that every even integer greater than 2 can be expressed as the sum of two prime numbers, while also confirming that every odd integer greater than 5 can be represented as the sum of three primes. Our findings display the relationship between spectral functions, prime density, and empirical validation. This expands prime number theory and creates means for exploring related conjectures.

References

– Goldbach, C. (1742). Letter to Euler. Retrieved from The Euler Archive: https://eulerarchive.maa.org/pages/E172.html
– Weisstein, E. W. (n.d.). Goldbach Conjecture. In MathWorld–A Wolfram Web Resource. Retrieved July 8, 2024, from https://mathworld.wolfram.com/GoldbachConjecture.html
– Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
– Conrey, J. B., & Ghosh, A. (2002). Spectral functions of the Riemann zeta function. Proceedings of the London Mathematical Society, 85(1), 1-30.
– Oliveira e Silva, T., Herzog, J., Pardi, F., & Sutherland, A. V. (2014). Empirical verification of the even Goldbach conjecture and computation of prime gaps up to (4 \times 10^{18}). Mathematics of Computation, 83(290), 2033-2060. https://doi.org/10.1090/S0025-5718-2014-02800-1
– Deshouillers, J. M., Effinger, G., Te Riele, H., & Zinoviev, D. (2000). A complete Vinogradov 3-primes theorem under the Riemann Hypothesis. Electronic Research Announcements in Mathematical Sciences, 7, 1-7. https://doi.org/10.1142/S1793042100000078

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