Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's *Street Fighting Mathematics*. In summary, the book used a integral estimation heuristic from spectroscopy

$$\int_{\mathbb{R_{\ge 0}}} f(x) dx \approx \max(f) * (\text{point where}\ \frac{1}{2} \max(f)\ \text{is achieved}) $$

to estimate the Gamma function with $f(x) = f_t(x) = x^{t}e^{-x} $. This leads to the estimate

$$\Gamma(n) = \int_{\mathbb{R}_{\ge 0}} x^{n}e^{-x} dx \approx \sqrt{8 n} \left(\frac{n}{e}\right)^n$$

which is an extremely good estimate (the "proportionality constant" $\sqrt{8}$ is correct to within 10% with correct order of growth.) This heuristic was very helpful in understanding the growth of the actual formula $\Gamma(n) \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$.

I think approximations of this sort is useful because

- It gives a sense of what the answer "ought" to be.
- When the approximation deviates from the actual answer, it's interesting to think about which part of the approximation failed.

Another "back-of-envelope calculation" is the calculation for the Prime Number Theorem in Courant and Robbins, *What is Mathematics?*

**My Question.** I am looking for similar instances in mathematics where "back-of-envelope calculations" such as the above leading to good intuition in mathematics.

For the purpose of my question, let's require that the calculation addresses questions in pure mathematics (so, no physics, engineering, etc. since there seems to already be plenty of literature on this).

Edit: as per helpful feedback from Peter LeFanu Lumsdaine, I removed two requirements: "Does not require anything beyond, say undergraduate mathematics" and "Does not formalize into a rigorous proof."

Edit 2 (as per helpful discussion in the comments): part of what I am interested in is *how* people use various techniques to compute/approximate objects of interest. For instance, I think we can all agree that the use of integral approximation demonstrated above is quite creative (if not, a nonstandard way of approaching Stirling). In response to Meow, topological invariants for "similar" (homotopy equivalent, homeomomorphic, etc) mostly amounts to the "same sort" of argument, so I would count that as "one" approximation argument unless there is a particular example where the heuristic argument is highly nontrivial.

increasethe number of undergrad-friendly answers you get. And “does not formalise into a rigorous proof” — I take the point of the questions you want to avoid overlap with, but good intuitive heuristics often do at least provide a significant insight towards a proof. (And when no proof based on the heuristic can be found, that may suggest the heuristic is wrong in general, and only gave the right answer by chance.) $\endgroup$ – Peter LeFanu Lumsdaine Nov 28 '20 at 10:13