Small results in number theory, written up by a random Reddit user.
A simple calculation with consecutive integers gives rational numbers approaching π: add the distances from each integer to the others, take reciprocals, add them, and multiply by the size of the group. I explain the limit using a Riemann sum and a picture of a right angle. The same picture shows that every approximation is larger than π.
For p-adic Moy-Prasad subgroups of GLn above depth 1/(p−1), I give an exact uniformity test. Logarithms supply the roots, and commutators do the rest. Depth one for odd primes and two for p=2 are sharp universal bounds, as 3×3 examples show. Uniformity can come and go as the depths grow.
The Association for Human Mathematics lists 809 mathematicians who commit to not publishing texts generated by AI, and 532 of them state that they do not use AI models in their research at all. I compare the arXiv output of these mathematicians with that of all other authors of arXiv mathematics papers, following an analysis plan that was fixed before any result was computed. From the five years before ChatGPT to the two years after the first reasoning models, the output per author of the AI-free group grew by 15%, almost exactly as much as that of comparable mathematicians, whose output grew by 17%. Only in the summer of 2026, when the number of all arXiv mathematics papers suddenly doubled, did the comparable mathematicians pull ahead, with an interval that only just excludes no difference. The AI-free group mentions AI tools less often, writes more coauthored papers than before and is closely connected by coauthorship.
I give a short proof of Bugeaud's S-part bound for integers with Sturmian digit strings. The argument uses the p-adic Subspace Theorem in dimensions two and three and also applies to finite words of linear factor complexity.
I study positive integers satisfying n = s(nk), where s is the decimal digit sum. A congruence condition and a random digit model explain patterns in the solutions. The model suggests infinitely many pairs and triples of consecutive exponents. I conjecture that 90 is the only n > 1 with four consecutive exponents. In other bases, an explicit construction gives arbitrarily long runs.
A number n is strongly pseudoperfect if some set of its divisors is closed under d ↦ n/d and adds up to 2n. I show that two such numbers a and b whose divisor sets contain 1 and n can be mixed into a2b, even when they are not coprime. This gives infinitely many primitive odd examples with only three prime factors, including ones of Euler's form, and so answers Questions 18 and 20 of Wang and Zelinsky. I also count these examples and give two shorter proofs of their results.
I write under the name TacoYaci, a random Reddit user who is a professionally trained mathematician in real life. Here I collect small results that I find fun, mostly in elementary number theory.