Demonstrating Goldbach's Strong Conjecture by Deduction using 4x ± 1 Equations in Loops and Gaps of 4

Abstract

Bahbouhi Bouchaib

After defining the writing of even numbers with equations 4x ± 1; the article shows that odd numbers obey loops of 4 numbers to infinity that obey the equations 4x ± 1. Each odd number in the loop can be prime (P) or composite (C), but for an even number E to be E = P + P' such that P' > P and such that P' > E/2 and P < E/2, P and P' must belong to two loops symmetrical with respect to E/2 and occupy the same positions in them and specific unit digits. The article shows that the three possible sums of an even number E are E = C + C'; or E = P + C; or E = P + P' (C is composite and P is prime). The article demonstrates that E = C + C' ↔ E = C + P ↔ E = P + P' ; and that one sum can be converted into another by substracting and adding gaps of 4n from or to the two terms of addition. This is a deductive demonstration of Goldbach's strong conjecture shown here for the first time.

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