A logical approach to validate the Goldbach conjecture: Paper 1/3

Abstract

This paper is a first of the series of three papers which provide a general proof to validate the Goldbach conjecture. This conjecture states that every even number can be expressed as a summation of two prime numbers. At the onset, concept of successive-addition of‐digits‐of‐an‐integer‐number (SADN) and its properties in terms of basic algebraic functions like addition, multiplication and subtraction are discussed. SADN classifies odd numbers into 3 sequences—the S1, S3 and S5 sequences—which comprise of odd numbers having SADN 7 or 4 or 1; SADN 3 or 9 or 6 and SADN 5 or 2 or 8 respectively. The S1 and S5 sequences are of interest in the analysis. Furthermore, composites on the S1 sequence are derived as products of intra-sequence elements of the S1 and S5 sequences while composites on the S5 sequence are derived as products of inter-sequence elements of the S1 and S5 sequences. SADN also shows why such combinations for even numbers of SADN (1, 4, 7) will be found on the S5 sequence while those for even numbers of SADN (2, 5, 8) will lie on the S1 sequence and both the sequences have a role to play in identifying the prime number combinations for even numbers with SADN (3, 6, 9). Thereafter, analysis moves to calculating the total number of combinations for a given even number that would include combinations in the nature of two composites (c1 + c2); prime & composite (p + c) and two primes (p1 + p2). Identifying the total number of c1 + c2 and p + c combinations yield the number of p1 + p2 combinations. The logic employed in present discussion shows that at least one such p1 + p2 combination exists for the even numbers having SADN digit within 1 to 9. This encompasses all even numbers and hence generalizes this method for all even numbers.