Quantum defect and common fractions

The common fractions N1/N2, where N1 and N2 - the small integers, quite often are used at the quantum-mechanical description of microcosm objects (for example, fractional charges of quarks and some quantum characteristics, such as particles spin). Recently the fractional quantum Hall effect was discovered, and common fractions have considerably expanded their presence in microcosm physics. The theory of the fractional quantum Hall effect has appeared nontrivial, so the Nobel Prize on physics in 1998 was awarded not only for discovery of the effect in 1982 (Daniel Tsui and Horst Störmer) but also for the theory creation in 1983 (Robert Laughlin). And now one more sensational discovery: common fractions were «detected» at the analysis of experimental characteristics of «hydrogen-like» atoms and ions (with only one electron on an outer shell). It has appeared, that the effective main quantum number of outer shell electron, that is, subject to quantum defect (Rydberg correction), can be expressed in common fractions.