I'm wondering if it makes sense to use the growth rate of a subset of $\mathbb{N}$ to approximate when an arbitrary structure or construction that is defined independently from the set is likely to occur i.e. for what sizes of values in the set.
For example, consider a magic square. Over $\mathbb{N}$, a magic square can be made with the first 9 values of the set:
The counting function for $\mathbb{N}$ is always about $C(x) = x$ (there are exactly x natural numbers less than or equal to $x \in \mathbb{N}$).
My idea was too see if we can do something similar for subsets of $\mathbb{N}$ where we know the counting function, or can at least approximate it well.
By PNT, the prime counting function is asymptotically equal to $C(x) = \frac{x}{ln(x)}$.
From what I've found, the smallest magic square with all prime entries is the following, found by Dudeney in 1917:
The entries are much bigger, and this is to be expected since primes grow slower and are less "dense" than naturals. Intuitively, this explains why it takes longer for primes to satisfy the arbitrary constraints of a magic square: there's a smaller chance that it will happen, so it happens later in the set of primes.
Now let's consider perfect squares. As far as I know, it is still an open question as to whether a magic square of squares exists. However, assuming it does exist, can we use the counting function for perfect squares which is about $C(x) = \sqrt{x}$ (slower-growing than primes) and what we know about magic squares over other sets to estimate when we should expect it to show up? I'm aware that this idea rests on the assumption that there is no hidden connection between the definition of a magic square and of primes and perfect squares, which may be unwarranted. However, if we assume that the first magic square shows up pseudorandomly, based only on a set's counting function/density/growth rate, are there standard methods for making a good prediction as to when this will happen for perfect squares? Is this something more advanced calculus can tackle? (By "more advanced" I mean anything I wouldn't have seen as a first year)
If such a prediction can be made, it could be useful for determining how likely it is that a magic square of squares exists given that it can't be made from the first $n$ elements of a set e.g. if an exhaustive search reveals no magic square of squares structure in the first 10 trillion perfect squares, how likely is it to still exist based on data from other sets with their respective densities?
*When I say magic square, I am talking specifically about a $3 \times 3$ magic square with distinct entries