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.

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 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.

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