The Mathematician: 12th August 2025

The 2025 International Mathematical Olympiad (rishimehta​.xyz). AI and mathematics intersect at the 2025 International Mathematical Olympiad, where OpenAI and Google DeepMind reported gold-medal performances, signaling a milestone in the AI Grand Challenge proposed in 2020. The contest featured six problems worth seven points each, with a gold cutoff of 35/42 and a record allocation of 72 gold medals due to many ties. OpenAI announced a gold-medal solution in natural language within 4.5 hours, delivering remarkably terse proofs and occasional self-encouragement phrases, prompting questions about verification and protocol. Google DeepMind followed with a similarly fast, externally vetted submission whose proofs appeared polished, almost indistinguishable from human work. Behind the scenes, ByteDance's Seed-Prover claimed gold on P1-P5 after a multi-day run, relying on a hybrid of formal and natural-language data, test-time library learning, conjecture generation, and scaling strategies. Harmonic, a lean-focused startup, also claimed gold with readable but still lengthy proofs; AlphaProof and AlphaGeometry from 2024 had come close, missing gold by a single point. The author reflects on why the Grand Challenge wasn't decisively solved, noting that fair grading in such settings requires external graders, invigilators, and robust verifier protocols, a point Terry Tao had emphasized. The piece surveys broader lessons: the pace of AI progress now shows energy and momentum across labs, with many viable approaches and strong path dependencies. Lean provers still matter, but the trend is toward building unified models that both reason and formalize, enabling agentic coding and faster progress on harder mathematical problems in the near future. This signals continuing AI math convergence

the MANIAC, or when machines strike deep inside enemy lines (am17an​.bearblog​.dev). Explores the lives of Paul Ehrenfest and John von Neumann, the rise of Artificial General Intelligence, and the philosophical implications of machine superiority

AI and ... (blog​.computationalcomplexity​.org). AI enhances navigation in Germany and engages in mathematical discussions; speculation on AI's capability to solve open math problems emerges

17 Year Old Hannah Cairo Found a Proof to a 40 Year old Problem (jeffmilner​.com). Hannah Cairo, 17, debunks the 40-year-old Mizohata-Takeuchi conjecture, challenging conventional mathematical intuitions about function behavior

How to solve a sliding puzzle (tanin​.nanakorn​.com). Solving sliding tile puzzles by tracking piece positions, using the border order as buffer, rotating borders, and comparing methods to BFS, A*, and Rubik’s cube

Mutually Attacking Knights (susam​.net). Susam Pal analyzes two knights attacking on an n×n chessboard, derives f(n)=4(n-1)(n-2) via Delta f(n)=8(n-2), Type A/B/C/D squares, attack degrees, and distinct-versus-identical knight counts, computation

Sangaku Puzzle I Can’t Solve (samjshah​.com). Sangaku puzzle discusses finding the radius of a small circle within a square, exploring geometric configurations using algebra and Geogebra

6 Impossible Puzzles With Surprising Solutions (mindyourdecisions​.com). Six challenging puzzles span maze coloring, bookworm geometry between encyclopedia volumes, liar puzzles with Alice Bob Charlie, forming squares from sticks, mapping keys, rhombus area

A thesis week (blog​.mitrichev​.ch). Mitrichev outlines randomized solving for problem H, uses edge-based backtracking, notes DP over DFS subtrees with at most backward edges, duplicating vertices for nonbipartite matching

Piecework (futilitycloset​.com). Henry Dudeney's Canterbury Puzzles (1907) asks how to cut an equilateral triangle into four pieces forming a square; 'strip of squares over triangles' offers proof

The creator and the machine (doc​.cc). Explores computer art's evolution from 1950s pioneers (Nees, Nake, Molnár, Laposky) to Sketchpad, generative design, mathematical perspective, Sangaku, Islamic patterns, Bauhaus rationalism, and AI authorship

Tensor computing from scratch part II - Advanced operations (e-dorigatti​.github​.io). Explores advanced tensor operations in a from-scratch PyTorch-like library: axis swapping (transpose), sub-tensor indexing, pretty printing, AbstractTenxor API, view/squeeze/unsqueeze, and iterative tensor traversal utilities essential

RNLA 2025 (mathsci​.ai). RNLA 2025 convenes IPAM workshop on Randomized Numerical Linear Algebra (Aug 11–15, 2025) featuring groups, travel/housing assistance; applications due Mar 31; led with Riley Murray

(BT) Diversity from (LC) Diversity (golem​.ph​.utexas​.edu). Explores relationships between Leinster–Cobbold and BT diversities, including maximum diversity's properties, subadditivity, complexity, and ecological applications in theoretical biology

Beam deflections by the conjugate beam method (leancrew​.com). Exploration of the conjugate beam method for beam deflection analysis, featuring Harold Westergaard's contributions and its application to structural engineering

Generating Gaussian Distributed Numbers Using JavaScript (jamesmccaffrey​.wordpress​.com). Generate Gaussian numbers using Box-Muller algorithm in JavaScript; discusses normal distribution, chi-squared tests, and includes a code implementation

Making the two-dimensional one-dimensional (johndcook​.com). Reducing 2D structures into 1D forms using concepts like postal codes, Traveling Salesman tours, and mathematical techniques like Hilbert curves

Thoughts on teaching multivariable calculus (blog​.evanchen​.cc). MIT 18.02 recitations emphasize writing polished notes, concrete examples: determinants, linear independence, normal vectors; plus sound-bites, repetition, flexible planning, and office-hours tweaks for student learning

Mathematical family (cameroncounts​.wordpress​.com). Reflections on mathematical mentorship, collaboration, and the concept of mathematical families involving notable mathematicians like G. H. Hardy, Paul Erdős, and more

A book you can count on (thonyc​.wordpress​.com). Benjamin Wardhaugh's 'Counting' explores the history and cultural significance of counting across various civilizations and time periods, combining anthropology with mathematical history

Harvey’s concept of circulation is as important as the concepts of Copernicus, Kepler, Newton, Darwin or Freud but not recognised as such (richardswsmith​.wordpress​.com). William Harvey's circulation theory parallels landmark concepts from figures like Copernicus and Darwin, yet its broader significance remains largely unrecognized

The Bejewled ‘Rubaiyat of Omar Khayyam’ at the Bottom of the Ocean (atlasobscura​.com). The jewel-encrusted 'Rubaiyat of Omar Khayyam' sank with the Titanic, showcasing the poet's journey from mathematician to literary sensation in the 19th century

A Thing or Two About RSA (nflatrea​.bearblog​.dev). Explains RSA basics: p and q primes, n, φ(n), e and d keys, M^e mod n, M = C^d mod n, attacks Wiener's and ROCA

Rough numbers between consecutive primes (terrytao​.wordpress​.com). Tao and Gafni resolve Erdős's question on rough numbers in prime gaps, using sieve theory, dyadic intervals, second-moment analysis to bound counterexamples and provide asymptotics

Knuckledragger Analysis Etudes (philipzucker​.com). Explores balls, open sets, and contraction via Z3-backed tactics, unfolding lemmas, axioms, and bagged intermediate goals to prove real analysis properties in a knuckledragger style

My Tom L post inspired a mathematical definition of Rabbithole (blog​.computationalcomplexity​.org). Lance Fortnow and Bill Gasarch discuss Tom L.'s post inspiring a mathematical Rabbithole, introducing RHS scores from time and enjoyment with Doomscrolling, binge-watching, and documentaries

Misleading plots of elliptic curves (johndcook​.com). John D. Cook analyzes misleading plots of elliptic curves over finite fields, contrasting real curves with mod p grids, misconceptions, Hasse bounds, and cryptographic implications

StepFun-Formalizer: Unlocking the Autoformalization Potential of LLMs through Knowledge-Reasoning Fusion (arxiv:cs). StepFun-Formalizer enhances autoformalization by integrating formal knowledge and reasoning through data synthesis, achieving state-of-the-art results with 7B and 32B models

On the intersections of nilpotent subgroups in simple groups (arxiv:math). Probabilistic proofs for Lisi-Sabatini and Vdovin's conjectures on nilpotent subgroups in simple groups, including applications to Sylow p-subgroups and group types

The Hardy--Ramanujan inequality for sifted sets and its applications (arxiv:math). Weighted Hardy–Ramanujan inequality for sifted sets; generalizes Halász, applies to shifted primes, Erdős multiplication table, divisors of shifted primes, Carmichael λ-image, ω(s(n)) normal order problems

Exact Polynomial Families Solving the Erdos-Straus Equation (arxiv:math). Construct four unbounded multivariable polynomials p1–p4 such that a = 4p_i(x,y,z)+1 yields explicit Erdős–Straus solutions; computational verification up to 10^9 covers all 1 mod 4

Decorrelation phenomena in mixed moments for orthogonal families of $L$-Functions and their applications: automorphic periods, Fourier coefficients of half-integral weight forms, and isotropy subgroups of Tate--Shafarevich groups (arxiv:math). Explores decorrelation in mixed moments of GL2 orthogonal L-functions under GRH, bounding quadratic-twist moments and applying to automorphic periods, half-integral Fourier coefficients, and Tate–Shafarevich isotropy

Cholesky decomposition for symmetric matrices over finite fields (arxiv:math). Develops Cholesky factorization theory for LPM matrices over finite fields, extending previous work and examining compatibility with Frobenius map and group operations

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