"Only divisible by itself and 1" is a darn elegant definition.
1, 2 and 3 are kind of special to me. In prime distribution studies, I discovered that they are special. It gets easier for some things if you consider primes only higher or equal to 5. Explaining distribution gets easier, some proofs become more obvious if you do that (tiny example: draw a ulam-like spiral around the numbers of an analog clock. 2 and 3 will become outliers and a distribution will reveal itself along the 1, 5, 7 and 11 diagonals).
Anyways, "only divisible by itself and 1" is a darn elegant definition.
When I was younger I had a period I often was thinking about prime numbers (before I got old and started thinking about the Roman Empire).
I noticed the same as you, and IIRC the (some?) ancient greeks actually had an idea about 1 as not a number, but the unit that numbers were made of. So in a different class.
2 and 3 are also different, or rather all other primes from 5 and up are neighbours to a multiple of 6, (though not all such neighbours are primes of course).
In base-6 all those primes end in 5 or 1. What is the significance? I don't know. I remember that I started thinking that 2*3=6, maybe the sequence of primes is a result of the intertwining of numbersystems in multiple dimensions or whatever? Then I started thinking about the late republic instead. ;)
If you work not only the primes, but also the modulus function value of each non-prime, things get even more interesting than thinking of base changes! To me, it reveals much more.
It's not entirely clear if that definition includes 1. On one hand 1 is certainly divisible by both itself and 1, but on the other hand they are the same number, so maybe it shouldn't count for "both", because the word "both" vaguely implies two distinct things. The usual "natural number with exactly two integer divisors" definition may not be as elegant but I think it is harder to misinterpret.
I see 1 as mostly an anchor. However, my thing is not about working out axioms and formal mathematics. I do some visualizations that can help demonstrate aspects of prime distribution.
I am fascinated by geometric proofs though. The clock thing is just a riff on Ulam's work. I believe there is more to it if one sees it as a geometric object and not just a visualization drawing. I could be wrong though.
1, 2 and 3 are kind of special to me. In prime distribution studies, I discovered that they are special. It gets easier for some things if you consider primes only higher or equal to 5. Explaining distribution gets easier, some proofs become more obvious if you do that (tiny example: draw a ulam-like spiral around the numbers of an analog clock. 2 and 3 will become outliers and a distribution will reveal itself along the 1, 5, 7 and 11 diagonals).
Anyways, "only divisible by itself and 1" is a darn elegant definition.