Two Theorems To Verify Goldbach's Strong Conjecture ; or To Refute it By an Uninterrupted Sequence of Composite Odd Numbers In The Interval [n-2n]

Abstract

Bahbouhi Bouchaib

Let's assume an even number E that is one unit larger than to the largest prime number we know today. Let's call this prime number Pl. Now we have 0—E/2—E and therefore Pl > E/2. For GSC to be true E must be sum of two primes P1 and P2 such that P1 < E/2 and P2 > E/2. Therefore we have to calculate E – P2 = X. If X is composite GSC is not verified; if X = P1 then GSC is verified. We then calculate E – P2 starting with Pl and all P2 till the one which is the closest to E/2. The question is: are all the Xs resulting from calculated E – P2 = X composites? Is it possible that all Xs might be composite which means non-verification of GSC? By contrast, if only one X is prime, then GSC is true. We see that GSC is much more likley to be verified in this case because a very long sequence of composite numbers is very unlikely to be continuous from E – Pl to E – P2 which is the closest to E/2. In other words there is at least one P2 prime in [ E/2—E] such that E – P2 = P1 → E = P1 + P2. The small primes are those that give the most composite numbers because their multiples are the most frequent but it is unlikely that all P2 of [E/2—E] would give composite numbers when calculating E – P2. If anyone, using this procedure, is able to find a sequence of composite numbers E - P2 = X in the whole E/2 – E interval; then He will be the first one who finds the solution to Goldbach's strong conjecture because this means its final rejection, and no mathematician can cast any doubt on his result. However, let us not forget that if only one X is prime, then GSC is true. As long as we cannot find this very precious and historical counterexample of an uninterrupted sequence of composite numbers by E – P2 = X (P2 is in E/2- E interval); Goldbach's strong conjecture will remain true although unprovable. This article gives the TWO THEOREMS of CONGRUENCE-MODULO which always predetermine whether GSC is true or not when calculating E - P2 = X. These two theorems described in this article will predict whether X is prime or composite. Nevertheless, these two theorems require the use of Euclidean divisions in series with the calculation of the remainders for each P2.

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