eng Science and Education Publishing American Journal of Mathematical Analysis 2333-8431 2025-12-17 11 1 12 13 10.12691/ajma-11-1-1 AJMA20251111 article Abhijit Manohar Corresponding author: armanohar0977@gmail.com 1 Kolhapur, Maharashtra, India 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. https://pubs.sciepub.com/ajma/11/1/1/ajma-11-1-1.pdf Goldbach conjecture prime numbers LeonhardEuler even integers tested for 4 x 10¹⁸