FRANKENSTEIN: A Computational Investigation of Large Wagstaff Numbers

This work presents a computational investigation of large Wagstaff numbers, with emphasis on the search for nontrivial factors, the characterization of surviving candidates, and the organization of a multistage computational workflow extending from factor search to probable-prime testing. For an odd prime exponent \(p\), the corresponding Wagstaff number is \(W_p=(2^p+1)/3\). Any prime divisor \(q\neq3\) of \(W_p\) satisfies the classical relation \(q=2kp+1\), together with \(2^p\equiv-1\pmod q\). The computational framework developed in this study uses this parametrization to restrict the factor search to admissible values of \(k\), followed by successive filtering, primality testing of the resulting \(q\), and exact verification of the divisibility condition. The investigation was conducted in successive experimental stages. Initial blind searches were expanded to larger \(k\)-ranges, followed by controlled population comparisons, survivor extraction, candidate prioritization, and deep searches. Across the validated historical catalogue, 69 distinct factor events were mathematically validated for 68 distinct prime exponents, including five independently verified factors with \(q\) in the \(10^{14}\)--\(10^{15}\) range. After searches through \(k\leq10^8\), 42 candidates remained without a detected factor in the investigated range. Two of these were subsequently extended to \(k\leq10^9\), where both remained without a detected factor. The work also separates factor detection from primality assessment. Survival of a candidate through a finite factor search does not establish probable primality, and probable-prime testing does not constitute a proof of primality. The resulting framework is therefore presented not as a new primality theorem, but as a reproducible computational workflow for exploring large Wagstaff numbers, recovering factors, reducing the candidate population, and selecting numbers for more expensive subsequent tests. Internal implementation details and proprietary optimization mechanisms are deliberately excluded from the public description.