Count the carries when you add two numbers in a prime-based system, and you know how often that prime divides a binomial coefficient.

This explores Kummer's theorem, a classical number-theory result: the number of times a prime p divides a binomial coefficient equals the number of carries you make when you add the two lower numbers in base p. To be direct, the collection has no note that explains or discusses Kummer's theorem. What follows is the standard result in brief, then the places where the corpus touches the same kind of mathematics.

The theorem itself is short. Take the binomial coefficient C(m+n, m), write m and n in base p, and add them by hand. Each time a column overflows and you carry a 1, you pick up one more factor of p in C(m+n, m). If there are no carries, p doesn't divide it at all. For example, in base 2, 3 + 5 is 011 + 101, which makes three carries, so 2³ = 8 divides C(8,3) = 56 = 8 × 7, and no higher power of 2 does. The reason is Legendre's formula, which counts the factors of p in a factorial using the digits of the number in base p. Kummer's theorem is what that count looks like when you apply it to a ratio of factorials. That framing, a ratio of factorials and the prime factors in it, is where the corpus comes in.

The closest material is the report that an AI system produced a formally verified Lean proof of Erdős Problem 728, a divisibility result about factorials with logarithmic gaps Did an AI system truly solve Erdős Problem 728 autonomously?. The note doesn't say which techniques the proof used. Still, questions about when one product of factorials divides another are exactly where digit-counting arguments like Legendre's and Kummer's are the usual tools. If you want to see what the elementary number theory behind these Erdős-style problems looks like in practice, that note is the one to open. The surprise in it is that the proof was checked by a machine before most people had read it.

That gap between a proof being verified and a proof being understood is the corpus's real theme here. Erdős problems became a test bed for AI partly because many of them, like divisibility questions, can be stated in a few lines but are hard to prove. Yet formal certification and human understanding are reported as separate outcomes, and understanding tends to lag behind Why did Erdős problems become a popular AI testing ground?. The Leiden Declaration answers this by keeping responsibility for correctness and credit with human authors. Its argument is that a proof does two jobs, establishing that something is true and conveying why, and formal verification only does the first Can AI-generated proofs ever replace human mathematical understanding?. Kummer's theorem is a good illustration of the second job. Knowing that p divides a binomial coefficient is one thing; seeing that the carries are what produce the factors of p is the understanding.

There is one more connection, and it is a cautionary one. Carrying digits in a given base is exactly the kind of step-by-step arithmetic that LLMs often get wrong. GSM-Symbolic found that model accuracy drops sharply when only the numbers in a problem change, which looks more like pattern matching than real calculation Does LLM math reasoning truly generalize or just pattern match?. That is why the successful math results in the corpus pair the model with an outside checker, such as Lean or numerical verification, rather than trusting the model's arithmetic Can opaque machine learning models help prove new mathematics?. If you want Kummer's theorem itself, a number theory textbook or a reference source will serve you better than this collection. If you want to know how AI is starting to work on problems built from these classical tools, the notes above are the place to start.