Cache systems fundamentally limit modern computing performance due to their inability to precisely capture data relationships. While achieving 85-92% hit rates, traditional systems rely on statistical heuristics that cannot guarantee relationship discovery, leading to suboptimal prefetching and resource waste. We present PFCS (Prime Factorization Cache System), which leverages the mathematical uniqueness of prime factorization to achieve deterministic relationship discovery with zero false positives. PFCS assigns unique primes to data elements and represents relationships as composite numbers, enabling the recovery of perfect relationships through factorization. A comprehensive evaluation across database, ML, and HPC workloads demonstrates an average performance improvement of x 6.2, 98.9% hit rates, and a 38% power reduction compared to state-of-the-art systems. The mathematical foundation provides formal guarantees impossible with approximation-based approaches, establishing a new paradigm for cache system design