The A3-A4 octave is the Prime Octave.
It is easier to use the words ‘Prime octave’ than to continually say “the A3-A4 octave”. So we started referring to the A3-A4 octave as the Prime Octave, and it stuck. It was specific and easy. The A3-A4 octave is the Prime Octave.
Same way with the A2/A3 octave. We refer to the A2-A3 octave as the Sub Prime octave.
It would have been nice, if Scientific Notation would have been an A to A thing. But it’s not, it’s C – C. If it was an A to A thing, we could simply refer to the 4th octave or the 3rd octave and everyone would know which notes we’re talking about. But technically the 3rd octave is C3 – B3, the 4th octave C4 – B4 and so on.
Calling A3-A4 the 4th octave is not technically correct. So we refer the A3-A4 octave the Prime octave and the A2-A3 octave the Sub Prime octave.