A3/A4 = The Prime Octave.
When talking or writing a lot about tuning, descriptive shortcuts are inevitable. Referring to the A3/A4 octave as the “Prime” octave came about as one of those shortcuts. Soon after that, the term ‘Sub-Prime’ was used to describe or identify the octave below the prime octave. The A2-A3 octave.
Those terms are not only used for the A3/A4 and the A2/A3 octaves but are also used for the intervals within those ranges. The ‘prime’ octave is referred to as both an interval and as a ‘range’ of notes (A3-A4) between and including A3 & A4. The ‘subprime’ refers to the interval A2/A3, and the range of notes (A2-A3) between and including A2 & A3.
It just makes it easier to talk about the prime 5ths which, for our purposes, are the lower prime 5th (A3/E4) and the upper prime 5th (D4/A4). The sub-prime 5ths being the A2/E3 (lower) and D3/A3 (upper).
When written, I try to use a distinction between the interval and the range of notes:
When a ” / “ is used between two notes, it is an interval. A3/A4 is the interval of an octave. The prime octave A3/A4.
When a ” – “ is used between two notes it is a range of notes. A3-A4 is the range of notes between and including A3 and A4.