Abstract: Two of the most widely used indices in geophysical research are the sunspot number as a measure of solar activity and the index Ap representing geomagnetic activity in the sub-auroral region. The sunspot number is a significant index of the solar activity because of its availability and reliability. Studies in solar terrestrial relationships have consistently shown the close and occasionally elusive link between the evolution of activity of the Sun and geomagnetic manifestations on the surface and in the magnetosphere. We use R/S assessing statistical singnificance and wavelet methods to assess statistical significance and confidence intervals of cross-wavelet phase and wavelet coherence. The Continuous Wavelet Transform is a common tool for analyzing localized intermittent oscillations in a time series. It is very often desirable to examine two time series together that may be expected to be linked in some ways. In particular, to examine whether regions in time-frequency space with large common power have a consistent phase relationship, causality between the time series is suggestive. Many geophysical time series are not normally distributed and we employed the Continuous Wavelet Transform. From two Continuous Wavelet Transforms we constructed the Cross Wavelet Transform which will expose their common power and relative phase in time-frequency space. We will further define a measure of Wavelet Coherence between two Continuous Wavelet Transform, which can find a significant coherence even though the common power is low, and show how confidence levels against red noise backgrounds are calculated. The rescaled range analysis (R/S) is proposed as a method to detect the correlations in pseudorandom number generators used in Monte Carlo simulations. In an extensive test, it is demonstrated that the R/S analysis provides a very sensitive method to reveal the hidden long-run and short-run correlations. Several widely used pseudorandom number generators are subjected to this test. In many generators, correlations are detected and quantified. In this paper, we used data of the Geomagnetic Ap index and the sunspot number from 1932 to 2009 to analyze the relationship between the Geomagnetic Ap index and the sunspot number by cross-wavelet method and the rescaled range analysis method. The results show that, 1) The Geomagnetic Ap index and the sunspot number have the similarity shape in the significant degree band with high frequency, and have a significant and steady oscillation period from 8.02 to 11.35 a; in the low frequency band, Geomagnetic Ap index and sunspot number have an uncontinuous oscillation period from 2 to 6 months, which is only a part of the year within the 95% of the red noise test. 2) There is significant in-phase resonance oscillation between the Geomagnetic Ap index and sunspot number in 8.02–11.35 a, in which the variation of the Geomagnetic Ap index occurs about 1.5 a after that of the sunspot number, and their phase relationship is steady. The Geomagnetic Ap index and the sunspot number also have relatively intermittent resonance periodicity from 4 to 6 months in the low frequency, but their phase relationship was not steady. 3) The Hurst index of the Geomagnetic Ap index and the sunspot number were 0.7923 and 0.8141, respectively. It indicated that the Geomagnetic Ap index and the sunspot number were continuous time series, and there existed a long persistence and chaoticity in the Geomagnetic Ap index and the sunspot number.