An Unconditional Partial Result Toward Erdős Problem #1059: Primes Avoiding a Positive Proportion of Factorial Shifts

This paper establishes an unconditional partial result toward Erd{o}s Problem #1059, concerning the infinitude of primes \(p\) for which \(p-k!\) is composite whenever \(k! 0\) such that, for every sufficiently large integer \(m\), there are \( m!/log(m!)\) primes \(p∈(2m!,3m!]\) satisfying \(p-k!\) composite for every \(1≤ k≤ cm\). More generally, for any integer-valued function \(K(m)=o(m)\), asymptotically almost every prime in this interval avoids all factorial prime shifts \(k!\) with \(1≤ k≤ K(m)\). The results are unconditional and require neither the Riemann Hypothesis nor any other unproved conjecture. Although the full Erd{o}s conjecture requires avoiding all factorial shifts up to \(m!\), these deductions establish simultaneous avoidance for a positive proportion of the relevant shifts. The arguments rely on established sieve-theoretic estimates and are presented as a rigorous partial result rather than a claimed resolution of the original problem.