Advances in International Applied Mathematics

This paper proposes a method to prove the Goldbach conjecture based on the combination of combinatorial mathematics and analytic number theory. By constructing a rigorous pairing system for odd sequences and introducing an improved estimation of prime distribution, we prove that every even number not less than 6 can be expressed as the sum of two odd primes. The main innovations are as follows: first, establishing a three-color pairing model, which rigorously classifies pairings into CC type, PC type, and PP type (where C denotes a composite number and P denotes a prime number); second, proposing a pairing density function φ(m) to quantitatively analyze prime distribution; finally, applying the improved Rosser-Schönfeld inequality to derive the precise contradiction regarding the invalidity of the conjecture.Theoretical analysis and numerical verification show that this method not only provides a graphical mathematical proof, offering new methodological support for exploring the underlying laws of prime distribution, but also presents new ideas for research on the applications of related number theory problems. These applications include the use of the conjecture in enhancing the encryption strength of big data, as well as in accelerating computations or improving accuracy in big data sequencing projects for DNA/RNA, among others.

Goldbach conjecture; Prime distribution; Combinatorial proof; Big data