The atomic number of an element represents the number of protons in the nucleus of its atoms, which uniquely identifies the element. In the case of phosphorus, its atomic number is indeed 15, as you mentioned.
However, it is important to note that the term "13th element" refers to a historical context known as the discovery order or the order of isolation of elements. This order is based on the chronological discovery or isolation of elements by scientists rather than their atomic number.
Phosphorus was historically considered the 13th element because it was one of the elements discovered and isolated by early chemists. Hennig Brand, a German alchemist, discovered phosphorus in 1669 through a process involving the extraction of phosphorescent material from urine. At the time, only a few elements were known, so phosphorus was the 13th element in the order of discovery.
With the advancement of scientific knowledge and the development of the periodic table, elements have been reorganized based on their atomic numbers. This reorganization provides a more systematic and accurate classification of elements, ensuring that the elements are ordered by their fundamental properties. As a result, phosphorus is now recognized as the 15th element based on its atomic number.