I was thinking about the Collatz conjecture a while back (I know, not the healthiest thing to think about). It occurred to me that while I might not be able to prove it true for *all* positive integers, I could spend 20 minutes and script up a simulation to generate a few million pseudorandom integers and test them to see if they follow the conjecture, and then publish my results as an empirical study such as is done every day in the physical and social sciences:

We tested ten million positive integers generated according to the pseudorandom algorithm defined by Jones [cite], then ran them against our TestCollatz(EnormousInt) function. We observed that 100% of the sample generated met the conjecture with a 99% confidence interval of plus or minus 0.1%. Conclusion: The evidence strongly supports the Collatz conjecture. We hope that future research in this area will be able to test billions or even trillions of integers to gain even more confidence in the accuracy of Collatz's model.

I soon realized that there was no algorithm to generate a statistically representative sample across a set of cardinality $\aleph_0$ with finite computing resources, and so my study idea was fruitless.

Stepping back from my misguided thought experiment, I realized I couldn't recall hearing of a recent *empirical* study in mathematics. Rather, the focus is on formal proofs and 100% confidence solely through logical and mathematical reasoning (not experimental data). Does empirical research exist or happen? I can find plenty of empirical research in mathematics *education* (e.g. checking whether starting children on arithmetic at age five rather than six raises standardized test scores), but mathematics education research is really a social science and not mathematics per se.

If there are recent *empirical* papers in mathematics, can you give me some sample citations of some interesting or notable ones to read? If empirical research is not done in mathematics today, why?

To be clear, I'm not asking about the Collatz conjecture specifically or why my idea was nonsense. I already know why. I'm asking if there are *good* examples of empirical studies in research-level math.

In terms of what I mean by "empirical study", I mean the standard "scientific method" of doing a literature review, forming a hypothesis, designing a study, randomizing experimental and control groups, doing the study, gathering the data, statistically analyzing it, forming a conclusion, and publishing. For example, if a medical researcher wants to show that a new drug works, they don't publish a five-page proof consisting of chemical reaction equations that they claim the drug triggers in the body, but rather they actually give the drug to a sample group and see what happens.

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