Ramanujan Challenge Results: AI-Assisted Evaluation Pipeline for Multi-Format Proofs
Alan Sorani, Ashvni Narayanan, Michael Shalyt, Tali Monderer, Rotem Kalisch, Hila Barkan, Elyasheev Leibtag, Shachar Weinbaum 等 11 位作者
AI 速览：分析Ramanujan挑战赛提交，比较AI模型、CAS与形式化工具的表现，并提出一种评估自然语言证明、CAS代码与Lean形式化的AI辅助流水线。
As AI's mathematical capabilities rapidly improve, meaningful benchmarks are shifting from university-level problems and math olympiads to research-level mathematics. The Ramanujan Challenge is a benchmark comprised of research-level problems in number theory, including unpublished results and open conjectures. It presented concrete formulas for mathematical constants whose correctness can be checked numerically, while their proofs may require non-obvious mathematics. Here, we analyze the Challenge submissions, measuring the relative success of different AI models, Computer Algebra Systems (CAS), and formal tools. We find correct submissions that are conceptually distinct from the official solutions, and a proof of one of the published open conjectures. Evaluating the submissions exposed a second challenge: judging complex proofs reliably at scale. We introduce an AI-assisted evaluation pipeline for natural language proofs, CAS code, and Lean formalizations that assesses semantic and logical correctness. Compared with the conventional one-shot LLM judgment, our pipeline reduces false-positive verdicts on natural language proofs from 23% to 6% and improves judgment accuracy on supporting code from 46% to 58%. This approach provides a methodology for evaluating research-level mathematics at scale, with potential applications across mathematics and AI for science.
The simple continued fractions for the Golden & Silver means are well-known. It is astonishing that, as far as we know, no one has published half-iterates (let alone quarter-iterates) for the corresponding algorithms. We also examine the cosine and logistic maps (with parameter $2 < λ< 3$).
Expander graphs have been central objects of study for over fifty years. Ramanujan graphs attain the asymptotically optimal nontrivial adjacency-eigenvalue threshold for infinite fixed-degree families. The original arithmetic constructions by Lubotzky-Phillips-Sarnak (1988) and Margulis (1988), and the iterative $2$-lift method of Bilu-Linial (2006) provide different approaches to Ramanujan graph constructions. Motivated by work on Mealy automata, we combine both approaches to give the following new arithmetic construction. For every prime $p\equiv 1\pmod {17}$, we construct a infinite tower of $(p+1)$-regular non-bipartite Ramanujan graphs on $2^n$ vertices, with label-preserving $2$-lifts between consecutive levels. These construction is strongly explicit: for fixed $p$, a neighbor or lift sign is computable in $O_p(n^3)$ bit operations. The construction uses fixed algebraic $2$-adic transformations, specified by finite data computable in preprocessing depending only on $p$. The main number-theoretic tool is Blasius's Ramanujan-Petersson theorem for Hilbert modular forms, applied through the Jacquet-Langlands correspondence to obtain the sharp bound $|λ|\leq2\sqrt p$ for every nontrivial adjacency eigenvalue of our $(p+1)$-regular graphs. The transformations preserve $2$-adic distance, which gives the binary covers and an explicit formula for their signings.
We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let $C_\mathcal{ L}(D,G)$ be such a code over $\mathbb F_q$, with length $n$ and dimension $k$. Assume that $3\leq k\leq n-3$. We prove that $n<(2q+10\sqrt q+16)/3$. This bound holds for every prime power $q$ and for both parities of $k$. It gives a uniform upper bound of $2q/3+O(\sqrt q)$ for the maximal length. The explicit bound is strictly smaller than $q+1$ when $q\geq125$.
We study the complexity of two fundamental problems attached to p-adic lattices: computing an N-orthogonal basis, and deciding whether a given basis is N-orthogonal. When the norm is supplied through an explicit diagonal representation, both problems are polynomial. However, in the black-box model, where the ultrametric norm N is fixed and only accessible through an evaluation oracle, we prove that both problems require $Ω$(p^{v-1} ) queries in the worst case, where v $\le$ n is the largest multiplicity of a commensurability class of N and n is the dimension of the ambient vector space. We then analyse the optimality of these bounds by providing explicit algorithms. Our results answer formally an open question of Zhang, Deng and Wang and refine the recent results of Duan and Wu by providing pointwise query bounds depending on the norm's structure, in addition to uniform bounds.
Every semi-magic square of order $n$ satisfies the axial moment identities $\sum_{i,j}s_{ij}j^m=\sum_{i,j}s_{ij}i^m=M(n)\sum_jj^m$ for all $m\ge1$, where $M(n)=n(n^2+1)/2$. For real matrices the converse requires $1\le m\le n-1$, whereas for arrangements of $\{1,\dots,n^2\}$ fewer moments suffice once $n\ge14$. We define the permutation threshold $k^*(n)$ as the least $k$ such that every arrangement satisfying these identities for $1\le m\le k$ is semi-magic. In terms of the column-sum deviations $γ_j$, the degree-$k$ conditions state that $\sum_jγ_jt^j$ is divisible by $(t-1)^{k+1}$, linking $k^*(n)$ to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives $k^*(n)\le\sqrt{2(n-1)\log(n^4/4)}$, we prove the converse realisability bound $k^*(n)\ge(\sqrt2-o(1))\sqrt n$ (and $k^*(n)\ge(\sqrt6-o(1))\sqrt n$ when $4\mid n$) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that $k^*(n)=n-1$ for $3\le n\le13$, $k^*(n)=n-2$ for $14\le n\le17$, and $k^*(n)\le n-3$ for $n\ge18$, and determine $k^*(n)$ for $18\le n\le40$ with computer assistance. Finally, at the $k=1$ level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy $E_{\mathrm{full}}$, for the four-term energy $E_4$ combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that $|L_4|=14{,}639{,}440$ arrangements of order $4$ satisfy $E_4=0$.
The diagonals of a regular pentagon dissect it into eleven pieces and draw a pentagram; the dodecahedron dissected by the twelve face planes of its inner core gives Kramer's seven "elementary cells" of 1982. We define the stellation dissection of a golden regular polytope in any dimension as the dissection of the convex hull of its final regular stellation by the facet hyperplanes of the core, show that the core has phi^(-d) of the inradius of the hull for d = 2, 3, 4, and carry the dissection out for the first time on the 120-cell. The foundation is the arithmetic of the golden ring Z[phi], the ring of integers of Q(sqrt 5): vertices and hyperplanes live in Z[phi]^d, every orientation test is the exact sign of an element of Z[phi], decided with integers, and the pieces are named by sign vectors. We develop this foundation from scratch (conjugation, norm, units, the primes sqrt 5, 2, 3, the exact sign test, the lattice x -> (x, conj x), the vertex modules of the dodecahedron and the 120-cell, the icosian ring) and let it carry the geometry. The definition bifurcates in dimension four: the star cut at depth phi^(-4), a unit, has 2,455,201 cells in 338 orbits of H4; the hereditary cut at depth 1 - phi^(-4) = sqrt 5 / phi^2, a unit times the ramified prime, induces the three-dimensional dissection on every facet and has 8,161 cells in 8 orbits. New here: all vertices of the dodecahedral and of the hereditary dissection lie in the vertex module of the polytope being cut, whereas the star cut needs the denominators 2, 3 and 5; the seven kinds of pieces of the dodecahedron are exactly the I_h-orbits of sign vectors, with stellation layer equal to the number of plus signs. We catalogue the four-dimensional pieces and treat the circumscribed disc, ball and 4-ball. Every number quoted is recomputed exactly by a script whose output is reproduced in an appendix.
For primes $p$, let $c(p)=\frac{\varphi(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$. Let $μ_G$ be its classical shifted-totient limiting law over primes and $μ_f$ its image under $x\mapsto-\log x$. Both laws have Hausdorff dimension zero, packing dimension one on every positive-mass Borel set, and almost-everywhere lower and upper local dimensions $(0,1)$. The four totient and divisor-sum laws over integers and shifted primes, and their logarithmic images, have lower and upper Shannon information dimensions in $(0.5503615,0.570489495)$. For convolution powers $μ_f^{*k}$, both information dimensions tend to one as $k\to\infty$, while each finite power has Hausdorff dimension zero. $μ_G$ gives full mass to absolutely normal Liouville numbers. Both Fourier transforms are $O((\log|τ|)^{-1/3}(\log\log|τ|)^{1/3})$. The empirical laws of $c(p)$ and $\varphi(p-1)/(p-1)$ for $p\le x$ have sharp Kolmogorov order $1/\log x$ in fixed reduced progressions; over unrestricted primes, they and their logarithmic images converge in $1$-Wasserstein distance at rate $O_B((\log x)^{-B})$ for every fixed $B>0$, possibly with ineffective constants. For $μ_f$, we obtain concentration expansions to every fixed inverse-logarithmic order, quantitative maximizer localization near $\log3$, and Fourier resonance bounds. For $U_T$ uniform on $[0,T]$, $(\log T)\widehat{μ_f}(U_T)$ converges in distribution to a circularly symmetric random Euler product, with convergence of modulus moments for every fixed real $s>-1$. Strongly additive functions with prime values $(\log\ell)^{-A}$ have absolutely continuous integer and shifted-prime laws for $A>0$. We give certified number-theoretic transform prime searches. A lattice appendix determines second and third distinct nonzero squared norms of type-$A$ tensor products and gives cyclotomic smoothing bounds. These results are unconditional.
This monograph develops finite field theory through root sets, Frobenius orbits, trace duality, and cyclic modules, with applications to polynomial constructions and algebraic coding theory. Root-set descent and base extension are expressed through finite reduced algebras, while linearized polynomials connect normal bases, invariant subspaces, and linear recurrences. Coefficient pairings and derivative formulas recover information lost by trace pairings in quotient algebras with repeated factors. Cyclotomic cosets are studied through additive stabilizers, power substitution, and compatible factor lifting, including extensions to Galois rings. A central theme connects low-degree representations of subfields with conjugate root blocks, polynomial composition, and recovery from relative trace data. Return maps on root fibers determine factor degrees and organize constructions of irreducible polynomials, affine symmetries, and permutation polynomials. Applications include cyclic and constacyclic codes, BCH and Goppa distance bounds, rank-metric codes, trace codes, and Fourier support. Character orthogonality and Gauss and Jacobi sums yield counting formulas, a proof of the primitive normal basis theorem, and recurrences for extension-field character sums. Gaussian periods connect these methods with short linear relations in multiplicative subgroups. Proofs of Weil bounds for polynomial character sums combine auxiliary polynomials with generating polynomials and estimates for their reciprocal roots. The emphasis throughout is on explicit arguments connecting field structure, orbit structure, and linear dependence.
Permutation polynomials over the Galois ring $\mathrm{GR}(2^k,m)$ are nonlinear components for symmetric primitives that add keys in the ring, as in secure computation modulo $2^k$. Rønjom and Sandrib showed that their differential uniformity is at least $2q^{k-1}$, where $q=2^m$. For the class studied here it is attained on top-layer differences, while almost all differences are units. We ask how small the difference table can be on unit differences when the residue permutation is fixed and only higher coefficient digits of its lift are chosen. Let $m\ge5$ be odd, $k\ge3$, and let $p$ be a reduced quadratic almost perfect nonlinear (APN) permutation of $\mathbb{F}_q$ whose derivative is nowhere zero and two-to-one. Over all lifts of $p$, the least possible maximum on unit differences is $q$ or $2q$, decided by two invariants of $p$ and the level $k$. We describe every lift attaining it and show that the coefficients modulo $8$ decide optimality at every level. For every odd $m\ge5$, residue permutations with at most $16$ monomials attain $q$ after one $2$-adic digit of one coefficient of the natural lift is changed. For an explicit infinite family, one added monomial lowers the maximum on unit differences from $4q$, or from at least $q^2$, to $2q$ at every level without raising the degree. Under a uniform starting state and independent uniform round keys, valuation preservation yields layerwise bounds for characteristics and two-round differentials that distinguish lifts with the same global uniformity. Core results are formalized in Lean 4.
We study completely normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, that is, elements that are normal over every intermediate field, through partial Lagrange resolvents. In these coordinates, complete normality splits into independent components, in a way closely related to Hachenberger's decomposition theory. Our main tool is a structure theorem for the components; together with a finite computation, it shows that every component has weight (the sum of the probabilities that its conditions fail) at most $3/4$, uniformly in $n$. Combined with the classical character-sum method, this gives an explicit sufficient criterion for the existence of a primitive completely normal element. We verify the criterion for all $n>q$ except $70$ pairs $(q,n)$, which we settle with explicit elements found by computer. Since the case $n\le q$ is due to Garefalakis and Kapetanakis, this confirms the conjecture of Morgan and Mullen (1996) that every $\mathbb{F}_{q^n}$ contains a primitive element that is completely normal over $\mathbb{F}_q$. The same estimates give two further results. First, Hachenberger's conjectured lower bound $(q-1)^{n'}q^{(p^a-1)n'}$ for the number of completely normal elements, where $p$ is the characteristic and $n=p^an'$ with $p\nmid n'$, holds for all pairs $(q,n)$; it was known for regular pairs. Second, primitivity and complete normality are asymptotically independent: as $q^n\to\infty$, the number of primitive completely normal elements is $(1+o(1))\,\varphi(q^n-1)/(q^n-1)$ times the number of completely normal elements (for fixed $n$ this was known). Our arguments for $n>q$ and for both further results, including every computer-assisted step, are formalised in Lean 4 with Mathlib; the formalisation takes the case $n\le q$ and an explicit bound of Rosser and Schoenfeld as axioms and relies on Lean's compiler and interpreter for the large computations.
We prove two universality theorems for neural networks consisting of a single neuron. First we prove a strong form of the Bovdi--Geche universality conjecture on neurons over finite fields, where the activation function is a certain modification of the discrete logarithm, using the Weil bound on multiplicative character sums over finite fields. Then we show a generalisation of Aizenberg's universality theorem for complex binary neurons with a weight vector independent of the target function, positively answering another question in the literature. The proof of the latter uses an equidistribution theorem for the dilates of certain maps into tori.
Hofstadter's G function is recursively defined via G(0) = 0 and then G(n) = n-G(G(n -1)). Following Hofstadter, we vary the number k of nested recursive calls in this equation and obtain a family of functions (F\_k). In a previous article, we proved that this family is ordered pointwise: for all k and n, we have $F_k(n) $le$ F_{k+1}(n)$. Here we propose a much shorter proof of this theorem, without infinite morphic words. The key ingredient for this direct proof is a new formula expressing iterates of the preimages of $F_k$ . Thanks to this formula, we also prove several conjectures mentioned in our previous article. In particular we obtain that $F_k(n) < F_{k+1}(n)$ when n > (k + 1)(k + 6)/2. We provide a certified formalization of all these results in the Rocq proof assistant.
Minjia Shi, Xuan Wang, Bouazzaoui Zakariae, Jon-Lark Kim, Patrick Solé
Wall-Sun-Sun primes (shortly WSS primes) are defined as those primes $p$ such that the period of the Fibonacci recurrence is the same modulo $p$ and modulo $p^2.$ This concept has been generalized recently to certain second order recurrences whose characteristic polynomials admit as a zero the principal unit of $\mathbb{Q}(\sqrt{d}),$ for some integer $d>0.$ Primes of the latter type we call $WSS(d).$ They correspond to the case when $\mathbb{Q}(\sqrt{d})$ is not $p$-rational. For such a prime $p$ we study the weight distributions of the cyclic codes over $\mathbb{F}_p$ and $\mathbb{Z}_{p^2}$ whose check polynomial is the reciprocal of the said characteristic polynomial. Some of these codes are MDS (reducible case) or NMDS (irreducible case).
Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{η_1,\ldots,η_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $η_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{η_i,η_j\in S:η_i\neqη_j}|\mathcal N(η_i-η_j)|\] where $\mathcal N$ denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.
Tony Feng, Trieu Trinh, Garrett Bingham, Jiwon Kang, Shengtong Zhang, Sang-hyun Kim, Kevin Barreto, Carl Schildkraut 等 24 位作者
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.