Short Proofs in Combinatorics and Number Theory
This presentation explores three elegant results in arithmetic combinatorics and analytic number theory, each addressing questions inspired by Paul Erdős. The paper establishes polylogarithmic bounds for prime divisors in binomial coefficients, constructs an additive basis that resists syndetic partition, and proves that sequences of prime multiples are never well-distributed. Remarkably, all proofs were generated by an AI language model, demonstrating the emerging capability of machine reasoning in mathematical discovery.Script
A language model just solved three problems that trace back to Paul Erdős—problems about primes hiding in binomial coefficients, about sumsets that refuse to be tamed, and about sequences that cluster when they should spread. Every proof in this paper was written by AI.
The authors tackle three distinct challenges. First, they bound how far we must look in Pascal's triangle before small primes dominate. Second, they construct a set where every partition breaks regularity. Third, they prove that multiplying primes by any real number produces clustering, never uniform distribution.
The first result reveals surprising order in the chaos of binomial coefficient factorization.
The authors prove that for sufficiently large n, some binomial coefficient with k around 6 times log n squared has small prime divisors whose product exceeds n squared. The proof is elementary, avoiding deep theorems about prime gaps. A matching lower bound construction shows this polylogarithmic growth is nearly tight.
The second result exposes limits on how neatly we can color sumsets.
Burr and Erdős asked whether every additive basis of order two can be partitioned so both color classes remain syndetic under doubling. The authors construct a basis where every partition forces at least one color to develop arbitrarily large gaps. The design uses intervals covered only via unique pairings, ensuring rigidity under partition.
The third result completes a question Erdős posed about equidistribution. The authors prove that multiplying primes by any real number produces a sequence whose fractional parts cluster infinitely often. Using modern results on bounded prime gaps in progressions, they force residues to bunch modulo carefully chosen denominators, breaking the local regularity condition for all alpha.
Three problems, three short proofs, all discovered by machine reasoning. These results show AI can now navigate the terrain between combinatorics and number theory with creativity and rigor. Visit EmergentMind.com to explore more research and create your own video presentations.
