The 3x+1 problem is a problem of continuous iteration for integers. According to the basic theorem of arithmetic and the way of iteration, we derive a general formula for continuous iteration for odd integers. Through this formula, we can construct a loop iteration equation and obtain the result of the equation: the equation has only one positive integer solution. In addition, this general formula can be converted into a linear indeterminate equation. The process of solving this equation shows that the relationship between the iteration result and the odd number being iterated is linear. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question.

Initial water content significantly affects the efficiency of soil stabilization. In this study, the effects of initial water content on the compressibility, strength, microstructure and composition of a lean clay soil stabilized by compound calcium-based stabilizer were investigated by static compaction test, unconfined compression test, optical microscope observations, environment scanning electron microscopy, energy dispersive X-ray spectroscopy and X-ray diffraction spectroscopy. The results indicate that as the initial water content increases in the range studied, both the compaction energy and the maximum compaction force decrease linearly and there are less soil aggregates or agglomerations, and smaller proportion of large pores in the compacted mixture structure. In addition, for specimens cured with or without external water supply and under different compaction degrees, the variation law of the unconfined compressive strength with initial water content is different and the highest strength value is obtained at various initial water contents. With the increase of initial water content, the percentage of oxygen element tends to increase in the reaction products of the calcium-based stabilizer, whereas the crystalline mineral of the soil did not change obviously.