The enormous popularity of finding the natural solutions of the Fermat–Pythagorean equation x2 + y2 = z2 has given rise to a large number of works devoted to finding formulas that allow one to directly write out solutions of the equation. Nevertheless, the magic contained in this problem makes us refer to it again and again. Another such case is described in this paper. The starting point is the formula for the difference of squares. It turns out to be sufficient to obtain a rather interesting result: if we take any natural odd number, square it, and consider the resulting number as the sum of two consecutive natural numbers, these numbers together with the original number give the Fermat–Pythagorean triplet. The proposed work is devoted to the proof of this and other more complex and interesting statements.