Short proofs in combinatorics and number theory

Abstract: We give a triplet of short proofs, each of which answers a question raised by Erdős. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i + A_i$ has bounded gaps, and the final concerns whether ${αp}$ is "well-distributed" in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI.

• The paper shows that the minimal k for which small prime factors in binomial coefficients exceed n² is bounded by O((log n)²) with explicit constants and matching lower bounds.
• It constructs an additive basis of order two whose partition always yields a non-syndetic sumset, addressing a problem posed by Erdős and Burr.
• The work proves that sequences {αpₙ} are not well-distributed, using bounded prime gap results to fully resolve an Erdős problem on distribution.

Short Proofs in Combinatorics and Number Theory

Overview

The paper "Short proofs in combinatorics and number theory" (2603.29961) addresses three questions inspired by Erdős, establishing results in arithmetic combinatorics and analytic number theory. Each result features a minimalistic proof and, notably, all proofs are attributed to an internal OpenAI model. The addressed questions pertain to (i) growth of small prime factors in binomial coefficients, (ii) syndeticity in sumset partitions of additive bases, and (iii) the well-distribution of sequences $\{\alpha p_n\}$, with $p_n$ denoting the $n$th prime.

Polylogarithmic Thresholds for Binomial Coefficient Prime Divisors

The first result concerns the function

$$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$

and evaluates

$f(n) = \min \{ 0 \leq k \leq n : u(n,k) > n^2 \}.$

The paper derives that $f(n)$ admits asymptotically polylogarithmic upper bounds:

$$f(n) \le \left(\frac{24}{\pi^2-6} + o(1)\right) (\log n)^2 \leq 6.20219 (\log n)^2,$$

for all sufficiently large $n$, while also constructing an infinite sequence $n_j \to \infty$ with

$f(n_j) \geq \left(\frac12 + o(1)\right) \log n_j.$

This improves upon prior bounds (e.g., $p_n$0 via analytic methods under strong prime gap assumptions) by providing bounds independent of deep results in the distribution of primes. The proof is elementary, relying on Legendre's formula and careful analysis of residue classes, together with prime number theorem considerations.

The findings demonstrate that, uniformly in $p_n$1, the multiplicative contribution of small prime factors within some binomial coefficient surpasses $p_n$2 for $p_n$3 of size $p_n$4. Further, the construction of $p_n$5 for which $p_n$6 is logarithmic shows the limit of possible improvement.

Additive Bases of Order Two Without Two-Syndetic Splittings

Addressing a problem of Burr and Erdős, the paper constructs a basis $p_n$7 of order two such that for all partitions $p_n$8, at least one of $p_n$9 or $n$0 fails to be syndetic (i.e., each has arbitrarily large gaps for at least one partition class).

The explicit construction of $n$1 leverages carefully spaced intervals and singleton elements. Critical to the argument is the design of intervals $n$2 that are covered only via unique combinations involving individual elements $n$3 and their associated intervals $n$4. This ensures that, upon any binary partition, the corresponding sumsets inevitably omit large intervals, thus destroying syndeticity for one color class.

This negative solution confirms that no general partition regularity principle applies for two-sumsets of additive bases of order two. The rigidity of the construction is reminiscent of earlier work by Erdős and Nathanson but refines the combinatorial design.

Non-Well-Distribution of $n$5

The third major result establishes that for all real $n$6, the sequence $n$7 is not well-distributed in the strong local sense introduced by Hlawka and Petersen. Specifically,

$n$8

fails for all $n$9.

The negative result is proved using results regarding bounded gaps between primes in arithmetic progressions (Maynard, Tao, Banks-Freiberg-Turnage-Butterbaugh). The argument applies Dirichlet's approximation theorem to engineer intervals of consecutive primes whose residues cluster modulo $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$0, and subsequently, their $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$1 fractional parts cluster into arbitrarily small intervals for infinitely many $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$2. This clusterization violates the local regularity criterion for all $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$3.

This extends earlier sporadic results, fully resolving Erdős's problem regarding well-distribution for all irrational $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$4, and, by the method, establishing the property for all real $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$5.

Implications and Future Directions

The results demonstrate the capability of AI-guided mathematical discovery in creative reasoning settings. The binomial coefficient threshold problem admits sharper description of the interplay between prime factorization structure and combinatorial quantities. The additive basis partition result underscores limits for partition regularity in sumset combinatorics and suggests that further structural analysis may demand higher-order or more quantitative invariants. The non-well-distribution result situates sequences of prime multiples into a unique category among classical equidistributed sequences and may have implications for pseudorandomness and diophantine approximation involving primes.

Future research directions raised by these results include:

• Tightening polylogarithmic constants for $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$6 or classifying $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$7 for which the logarithmic lower bound is sharp.
• Classifying additive bases of higher order with respect to partition syndeticity phenomena.
• Characterizing well-distribution properties of $$u(n, k) = \prod_{p \leq k} p^{v_p\left(\binom{n}{k}\right)}$$8 for other lacunary or structured sequences.

Additionally, the methodology—proof discovery via LLMs—presents an intriguing avenue for the exploration of short, conceptually clear proofs in classical problems, potentially enabling broader accessibility and new approaches to established conjectures.

Conclusion

This work provides concise, rigorous answers to longstanding problems at the intersection of combinatorics and number theory. The proofs, generated entirely by a LLM, highlight both the depth of contemporary AI reasoning tools and their emergent capacity to resolve nontrivial open questions. These results have direct implications on the understanding of prime factor thresholds in binomial coefficients, the combinatorial structure of additive bases, and the analytic distribution of prime-generated sequences.