Treffer: Prime Combinations of order n Hypothesis- PCn(Vô Hypothesis 3-VH3)

The PCn Hypothesis The PCn hypothesis extends classical conjectures about prime numbers, particularly the ternary Goldbach conjecture, by considering weighted sums of order \(n\) for prime triplets. For each odd prime \(p > 17\), the set of all triplets \((a,b,c)\) such that \(a+b+c=p\) is denoted \(\mathcal{S}_p\). The hypothesis posits that, for each triplet \((a,b,c)\) and each order \(n \ge 2\), there exist coefficient sets \(\{(a_1^{(i)},b_1^{(i)},c_1^{(i)})\}\) such that the sum of each class equals \(p\) and the weighted sum: $$q_n = a \prod_{i=1}^{n} a_1^{(i)} + b \prod_{i=1}^{n} b_1^{(i)} + c \prod_{i=1}^{n} c_1^{(i)}$$ is a prime number. An extended corollary—the **Full Correspondence Property**—asserts that each triplet in \(\mathcal{S}_p\) corresponds to at least one such \(q_n\). Experimental results for primes \(p = 19, 23, \ldots, 97\) and orders \(n\) up to 200 demonstrate the consistent existence of \(q_n\), even under the constraint of distinct coefficients, revealing an *“unusual stability”* phenomenon. No counterexamples were found, though some orders \(n\) required significant computational time. ## Mathematical Significance PCn advances the study of prime distribution by combining combinatorial and weighted product structures, offering new directions for combinatorial number theory, probabilistic modeling, and cryptographic applications. The hypothesis poses a significant theoretical challenge: understanding the deep structure of prime numbers through highly stable yet unpredictable combinations.