This paper presents a simplified joint synchronization scheme for the integer carrier frequency offset and physical cell identity using a primary synchronization signal (PSS) in 5G new radio (NR) communication systems. We demonstrate the efficiency of our proposed NR-PSS synchronization scheme by deriving its simplified implementation, which exploits the near-zero autocorrelation feature between cyclically shifted NR-PSS symbols. As a figure of merit, we compute the probability of detection failure of the proposed NR-PSS synchronization scheme and validate its accuracy via simulations. To illustrate the benefits and limitations of the proposed NR-PSS synchronization scheme, we compare it with the conventional NR-PSS synchronization scheme, considering factors such as detection performance and computation complexity. Numerical results indicate that, regardless of the channel environment, the proposed NR-PSS synchronization scheme achieves a significant reduction in arithmetic complexity while maintaining the same detection capability as existing NR-PSS synchronization schemes.