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\H{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 \(\gg m!/\log(m!)\) primes \(p\in(2m!,3m!]\) satisfying \(p-k!\) composite for every \(1\le k\le\lfloor cm\rfloor\). 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 \(1\le k\le K(m)\). The results are unconditional and require neither the Riemann Hypothesis nor any other unproved conjecture. Although the full Erd\H{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.