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Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems
Tony Feng, Trieu Trinh, Garrett Bingham, Jiwon Kang, Shengtong Zhang, Sang-hyun Kim, Kevin Barreto, Carl Schildkraut, Junehyuk Jung, Jaehyeon Seo, Carlo Pagano, Yuri Chervonyi, Dawsen Hwang, Kaiying Hou, Sergei Gukov, Cheng-Chiang Tsai, Hyunwoo Choi, Youngbeom Jin, Wei-Yuan Li, Hao-An Wu, Ruey-An Shiu, Yu-Sheng Shih, Quoc V. Le, Thang Luong
TL;DR
The paper asks how AI can help evaluate many conjectures marked “Open” when expert verification and literature searches are costly. It combines AI-based natural-language verification with human mathematical evaluation, finding that AI can identify some low-hanging results while leaving correctness, intended meaning, and novelty difficult to establish. The study therefore treats AI-assisted mathematics as promising for attention-bottlenecked discovery but requiring careful human oversight.
Problem
Evaluating hundreds of “Open” Erdős problems is difficult because expert assessment is scarce and the database status may not reflect the complete literature.
Method
Aletheia used AI-based natural-language verification to narrow 700 prompts to potentially correct candidates, which human mathematicians then evaluated for correctness and novelty.
Results
The study identifies some autonomous and literature-based resolutions, including a tentative autonomous resolution of Erdős-1051 among its reported positive results.
Takeaways & Limitations
AI may help harvest low-hanging results and accelerate attention-bottlenecked aspects of mathematics discovery if its reliability improves.
Abstract
from arXiv · showhide
We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.
1 Introduction
The paper uses AI verification and human mathematicians to examine problems marked “Open,” finding that many were technically correct but few meaningfully addressed the intended questions. The results also expose substantial difficulties in literature checking, novelty assessment, and interpreting problem statements.
- 1 Introduction: AI-based verification reduced 700 “Open” problem prompts to 212 potentially correct responses for human evaluation.Human mathematicians then assessed correctness, intended meaning, and novelty.
- 1 Introduction: 63 responses were technically correct, but only 13 meaningfully addressed the intended problem statements.The remaining technically correct responses were mathematically vacuous, ambiguous, or otherwise failed to capture the intended questions.
- 1 Introduction: Literature investigation was the most arduous stage because subtle formulation, notation, and definitional issues affected whether solutions were valid and novel.The number of meaningfully correct solutions fell during this investigation, primarily because of literature disentanglement rather than mathematical incorrectness.
- 1 Introduction: The 13 positive results comprised autonomous resolutions, partial AI solutions, independent rediscoveries, and literature identifications.The categories distinguish first correct solutions, first solutions to selected subquestions, rediscovered published results, and problems already resolved in the literature.
- 1 Introduction: The authors report four autonomous solutions but make no claims of novelty and note that some are comparable to student exercises given existing literature.They tentatively regard Erdős-1051 as a somewhat broader and mildly non-trivial case, while acknowledging a minor error in the original model output.
- 1 Introduction: Novelty classifications remain provisional because earlier human solutions may have been missed or may have existed implicitly as special cases of broader results.The authors describe their initial classifications as, at best, upper bounds on novelty.
2 Problems autonomously solved by AI
Aletheia found a first correct solution, as far as the authors could tell, while the resulting proofs required correcting cited exponents, constants, and minor logical errors. One representative argument combines a literature incidence theorem with a constructed family of circles to derive the target bound.
- Aletheia found the first correct solution, as far as the authors could tell, for problems in this set.
- The model’s cited theorem could not be located, while the analogous Pach–Sharir result has exponents (3/5, 4/5), not (2/3, 2/3).The final solution follows the model’s approach with corrected exponents.
- The report also corrects an unnecessary α_k + ϵ bound and a strict-inequality error in a lemma.The former could create dependency issues because n depends on ϵ; the latter was later formalised in Lean 4.
- The argument concludes α_k = Ω(k^1/4), establishing the desired result.The proof uses the incidence bounds and the limit argument to obtain this growth rate.
- The proof constructs circles centered at k selected points, with radii given by their distinct distances to the remaining point set.Each selected point contributes a family of circles, and every point outside the selected subset lies on at least k such circles.
- (n − k)k incidences provide the lower bound that is combined with the Pach–Sharir incidence theorem for circles.The theorem is applied with k = 3 and s = 2, yielding an asymptotic inequality after division by n and passage to the limit.
3 Problems with parts solved by AI
The paper presents AI-assisted solutions to several multipart Erdős problems, including a construction showing that bounded distinct distances can coexist with no four concyclic points. It also records scope limitations where only part of a problem was addressed or one question was omitted.
- Aletheia found the first correct solution to one of the multipart questions.
- One problem is treated only in part because the database also records a weaker formulation with an additional no-three-collinear-points assumption.The authors state that their solution fully addresses the formulation currently shown on ErdosProblems.com but not the weaker variant.
- The constructed set S has n=4m points on the coordinate axes, with 2m points on each axis and no four points on a circle.The construction uses sets P and Q on the y- and x-axes, respectively.
- Every point determines fewer than roughly 3n/4 distinct distances, so the answer to the corresponding question is negative.The construction gives at most approximately 3/4n distinct distances for each point, while avoiding four points on a circle.
- Distances from a point to its resident axis are integers, whereas distances to the orthogonal axis are irrational, so the two distance sets are disjoint.The proof handles points on either axis symmetrically and uses prime-power coordinates to rule out integer cross-axis distances.
- For the transfinite-diameter problem, Aletheia correctly answered the first question with an example, but its argument for the second question was incorrect and was omitted.The paper proves the first question by constructing two sets with the same transfinite diameter but different values of µ(F).
1. Construction of F1 (Positive Area) Let F1 = {0} ∪
The construction of F1 shows that a countable compact set with zero transfinite diameter can still force the associated lemniscate area to be at least π/4.
- F1 is contained in [0,1], so every monic polynomial with roots in F1 has its unit sublevel set containing the disk centered at 1/2 with radius 1/2.The triangle inequality gives |P(z)|<1 throughout that disk.
- In contrast, the F2 construction uses Q(z)=z(z−R), whose lemniscate area is at most 2π/(R^2−4) and tends to zero as R grows.For sufficiently large R, the paper obtains µ(F2)<π/4.
3. Conclusion
Because F1 and F2 both have transfinite diameter zero but yield different values of µ(F), the paper concludes that µ(F) is not determined by d∞(F.
- F1 and F2 both have transfinite diameter zero, while µ(F1)≥π/4 and µ(F2) can be arbitrarily close to zero.
- A human annotation notes that the transfinite-diameter claim was not justified or cited in the model output, although it follows from cited mathematical results and direct verification.
4 Independent rediscovery
The paper describes an AI-generated solution later classified as an independent rediscovery after a matching construction was found in an earlier comment. The underlying result establishes infinitely many solution pairs through an explicit family and a Pell-equation-based argument.
- The problem has infinitely many distinct pairs of disjoint finite sets of positive integers satisfying the required product identity.For every k≥3, the paper defines an explicit pair (A_k,B_k), with each k producing a unique solution pair.
- The proof verifies the product identity by substituting selected arguments into ratios of consecutive central binomial coefficients.
- The authors note that the argument is well-known to experts and connects the construction to standard Pell-equation facts about consecutive squarefull numbers.They also relate the third question to consequences of the ABC Conjecture.
- The proof of unbounded Q2(n_k+2) uses primes p≡5 (mod 8) and constructs k with n_k+2≡0 (mod p^2).Dirichlet’s theorem supplies infinitely many such primes, completing the divergence argument.
- The solution was reclassified as an Independent Rediscovery after van Doorn identified the same construction in a November 2025 comment on another Erdős problem page.The authors report that the logs showed no access to that page and that the comment postdated the model’s knowledge cutoff.
- For another problem, Aletheia gave an affirmative answer using a specific integer lattice designed to make squared distances integral and avoid relevant geometric symmetries.
1. Construction of the Lattice and Point Set
The construction embeds the ring of integers of Q(√−7) into R2 as a lattice, then selects the n points nearest the origin. A quadratic-form representation bounds the distinct distances, while the construction is verified to satisfy the required condition.
- Lattice construction: The ring OK of Q(√−7) is embedded into R2 to form the lattice Λ.The squared norm of m+kω is represented by the quadratic form Q(m,k)=m2+mk+2k2.
- Point-set construction: The point set Pn consists of the n lattice points closest to the origin.The construction places Pn inside a disk whose radius is controlled by the location of a lattice vector of size O(√n).
- Distance bound: Distinct squared distances in Pn are nonnegative integers represented by Q and bounded above by 4R2=O(n).Thus the number of candidate distances is controlled by represented values of the positive-definite form Q.
- Verification: The constructed Pn is verified to satisfy the required condition on the number of distinct distances.
3. Lower bound on the number of distances determined by 4 points
The proof excludes four-point configurations that could determine only two distances in the lattice. It rules out the relevant geometric configurations using irrational distance ratios and field-membership contradictions, completing the lower-bound verification.
- Configuration classification: Any four-point set determining exactly two distances must have one of several classified configurations, including an isosceles trapezoid, rhombus, or equilateral-triangle-based shape.
- Excluding configurations: The isosceles-trapezoid configuration is impossible because it requires an irrational squared-distance ratio incompatible with the lattice.
- Excluding configurations: An equilateral triangle cannot lie in Λ because its defining rotation introduces √−3, which is not contained in Q(√−7).
- Excluding configurations: A square cannot lie in Λ because its complex-coordinate description would require i∈Q(√−7), yielding a contradiction.
- Conclusion: Together, the preceding exclusions establish that the constructed Pn satisfies the desired four-point distance condition.
3. Case n ≥2
For n≥2, the argument bounds gd(n) by reducing it to the maximum size of an s-distance set and constructing a large binary-vector example embedded in Rd.
- Upper bound: The upper-bound argument uses the established Bannai–Bannai–Stanton bound for s-distance subsets of Euclidean space.
- Lower bound: A lower-bound construction uses binary vectors in Rd+1 with exactly s ones, all lying in a d-dimensional affine hyperplane isometric to Rd.
- Lower bound: For distinct vectors u and v, squared distances equal 2(s−k), where k is their number of common ones, so at most s distance values occur.
4. Calculating the Limit
The limiting calculation combines matching upper and lower bounds to obtain the asymptotic value of gd(n), while the section also records several examples where apparently open Erdős problems were resolved through existing literature.
- Limit calculation: For n≥2, the upper and lower bounds converge as d→∞ because their correction terms vanish, giving the limit 1/(n−1)!.
- Limit calculation: The resulting answer is 2 for n=1 and 1/(n−1)! for n≥2.
- Literature-resolved problems: Aletheia identified an existing Erdős–Newman result that gives a negative answer to one problem, with the construction producing a density-zero set not covered by a sufficiently sparse sumset.
- Literature-resolved problems: The partition relation ω^ω²→(ω^ω²,3)² follows from Schipperus’s characterization, with Larson’s stronger target-size result implying the stated case.
- Literature-resolved problems: Another problem has a negative answer because Berkes–Philipp proved a discrepancy lower bound that invalidates the proposed asymptotic bounds.
- Literature-resolved problems: The cycle anti-Ramsey problem was solved by Montellano-Ballesteros–Neumann-Lara, while the path problem was addressed by unpublished work of Yuan.
A Erdős-75: a case study within a case study
Erdős-75 illustrates that an apparently autonomous solution may be correct yet still require literature comparison and correction of the problem’s formulation.
- Case study: Aletheia produced a correct solution to Erdős-75 by identifying existing literature, while another model solved a strengthening of the stated problem.The strengthening was explicitly asked by Erdős, but the autonomous solution did not cite the relevant source.
- Formulation issue: The listed Erdős-75 formulation was not the intended one because Erdős’s formulation in one paper was flawed, despite being accurately transcribed.The intended formulation was accurately recorded in two other references.
- Related issue: Erdős-124 had a similar formulation problem, but its flaw was more obvious because the solution reduced trivially to known literature.The paper documents the autonomous solutions to both Erdős-75 and its strengthened version as originally listed.
A.1 Erdős-75
The Erdős-75 question is answered affirmatively by applying Lambie-Hanson’s construction, whose finite subgraphs have sufficiently slow chromatic-number growth to force large independent sets.
- Assessment: The paper notes that Aletheia’s proof is unnecessarily complicated: any f whose inverse is at most x^o(1) would suffice.The authors retain the double-exponential construction to reproduce the model’s proof faithfully.
- Construction: Lambie-Hanson’s theorem supplies a graph of chromatic number ℵ1 whose subgraphs with chromatic number at least k have at least f(k −3) vertices.The theorem is stated as proven in ZFC for every function f: N → N.
- Construction: Choosing f as a double exponential yields a graph where an n-vertex subgraph has chromatic number bounded by an iterated logarithm of n.The proof then compares this chromatic bound with the standard relation between independence number and chromatic number.
- Proof: The resulting chromatic bound implies an independent set exceeding n1−ϵ for all sufficiently large n, because n^ϵ eventually dominates the iterated logarithm.The proof introduces a threshold Nϵ beyond which the comparison holds.
- Conclusion: Yes, such a graph exists: every sufficiently large n-vertex subgraph contains an independent set larger than n1−ϵ for every ϵ > 0.The construction uses a graph of chromatic number ℵ1.
- Formulation: The correct statement should additionally require that the graph have ℵ1 vertices.This scope correction is stated before presenting the autonomous solution.
Erdős-75-strengthened-correct
The strengthened Erdős-75 formulation is answered affirmatively under the continuum hypothesis using a shift graph with chromatic number ℵ1 and linearly large independent sets in finite subgraphs.
- Problem and answer: The strengthened question asks for a graph of chromatic number and cardinality ℵ1 whose n-vertex subgraphs contain independent sets of size much larger than n.The paper states that its correct answer is affirmative.
- Construction: Under the continuum hypothesis, the constructed shift graph Γ(D) has chromatic number ℵ1.The graph is built from ordered pairs in a totally ordered set, with edges linking (x,y) to (y,z).
- Conclusion: The final theorem combines χ(Γ) = ℵ1 with the n/4 independent-set bound for every finite subgraph.The result is explicitly stated under the continuum hypothesis.
- Assumption: The proof depends on the continuum hypothesis, which is used to identify the relevant cardinalities and apply the Erdős–Rado argument.The hypothesis appears explicitly in the theorem statement and proof.
- Chromatic number: The coloring argument uses a bijection into binary sequences to obtain an upper bound, while the Erdős–Rado theorem rules out countable colorings.Together these auxiliary arguments establish the exact chromatic number.
- Independent sets: Every n-vertex subgraph has an independent subset of cardinality at least n/4.A random partition of the coordinate set produces an independent set whose expected size is n/4.