The invention of the Rivest–Shamir–Adleman (RSA) cryptosystem was a groundbreaking advancement in cryptography. While the RSA remains relevant in securing global communications and digital transactions, with widespread use in public parameter infrastructure (PKI) and secure online exchanges, its vulnerability to algebraic attacks must be addressed. In this paper, we propose an equation, thereby revealing its potential application in the factorization of the modulus $N$. By introducing this equation, we demonstrate a method in the first attack named the continued fraction for recovering the primes $p$ and $q$ without necessitating the original $\varphi (N)$ used in the RSA encryption. The results were extended to the case where a condition exists such that multiple sets of public parameters were used against a constant private parameter. We retrieved the primes $p_i^{\prime }s$ and $q_i^{\prime }s$ of the moduli $N_i$ via the lattice reduction technique. This breakthrough could potentially expose the prime factors while circumventing standard cryptographic barriers. Our findings open new possibilities for cryptographic analysis and challenge the presumed security of widely used RSA systems.