Perplexity
Perplexity develops AI tools for search, browsing, research, content creation, organization, and workflow automation, including perplexity.ai, Comet, and Computer.
Weighted Divergence Guarantees Infinitely Many Inhomogeneous Approximations
An OpenAI manuscript on the weak inhomogeneous Duffin–Schaeffer conjecture gives a sufficient condition for infinitely many shifted rational-approximation hits for almost every real number. For a fixed shift \(\gamma\) and non-negative tolerances \(\psi(q)\), the authors prove the result when \(\sum_q (\phi(q)/q)\psi(q)\) diverges, without requiring the approximating numerators to be coprime to their denominators. The proof’s main challenge is controlling the structured overlap among approximation intervals.
Infinitely Many Primes Have Squarefree Predecessors With Even Prime-Factor Parity
An OpenAI preprint proves that infinitely many primes p have p−1 squarefree with an even number of distinct prime factors. Its proof uses weighted candidates and analytic estimates to show that enough survive as primes after composites are removed; it does not claim that half of all primes have this property. Perplexity Computer adapted the preprint into an animated video.
Largest Prime Factors of Consecutive Integers Become Independent
An OpenAI preprint on the joint Dickman law argues that the normalized sizes of the largest prime factors of consecutive integers become statistically independent when integers are counted uniformly over an expanding range. In the limit, the chance that both largest prime factors fall below specified power thresholds is the product of the two familiar Dickman probabilities. The result also implies that each neighbor has the larger largest prime factor half the time, in natural density.
Intervals of Length O(k²/(log log k)²) Contain a Number Coprime to the Modulus
An OpenAI preprint on Jacobsthal’s function claims that if a modulus has \(k\) distinct prime factors, every interval of length \(Ck^2/(\log\log 3k)^2\) contains an integer coprime to it, for an absolute constant \(C\). The manuscript’s main argument establishes a positive count of survivors after small-prime exclusions, then shows that larger prime factors cannot eliminate them all. It does not give a numerical value for \(C\) or determine the bound’s optimal growth.
A Fontaine–Mazur Modularity Theorem at 2 Without Residual-Image Restrictions
The manuscript claims that, under its stated hypotheses, an odd, regular, two-dimensional Galois representation over the rationals with two-adic coefficients comes from a classical cuspidal eigenform, up to a Tate twist, without a restriction on its image modulo 2. It divides the proof into three stages: showing that the target eigensystem occurs in completed quaternionic forms, establishing classicality through a separate local argument at 2, and descending the resulting classical packet to the rationals.
Tame Hecke Eigensheaves Allow Arbitrary Unipotent Monodromy at Multiple Markings
An OpenAI preprint proves the existence of a nonzero perverse Hecke eigensheaf for \(SL_n\) bundles with flags at several marked points, when the associated \(PGL_n\) local system has Zariski-dense image and tame, unipotent monodromy at each marking. The theorem permits any unipotent local shape—not just regular unipotent monodromy—and preserves the full tensor and fusion structure of the Hecke operations. It requires no Frobenius structure.
A Claimed Zero-Free Region for Dirichlet L-Functions Begins at 11/12
An OpenAI preprint claims that every Dirichlet L-function, including the Riemann zeta function, has no zeros strictly to the right of Re s = 11/12—a fixed boundary that falls short of the Riemann hypothesis. Its proof, as presented in the manuscript, first bounds cancellation in a weighted sum over ideals in the Eisenstein integers, then uses a Mellin transform to turn that bound into a zero-free region. The key step is a deep mean-square estimate for a family of character sums; the paper’s account of that estimate is a roadmap rather than a full proof.
Quaternionic Division Points Are Finite Across All Indefinite Algebras
An OpenAI preprint proves that a reduced, irreducible, closed curve defined over the algebraic numbers and Hodge generic in the Siegel threefold contains only finitely many points whose abelian surfaces have an indefinite quaternion division algebra as their full rational endomorphism algebra. The finiteness holds across all such algebras, rather than separately for each fixed type. The proof establishes a height bound independent of the algebra, then rules out unbounded degrees by showing they would force the curve’s period image into a locus too small to contain it.
Prime-Degree Torus Packets Equidistribute Across Arbitrary Local Types
An OpenAI preprint proves that, in each fixed prime degree \(n \ge 5\), the volume-weighted measures on complete packets of arithmetic torus orbits converge to Haar probability as the discriminants of their multiplier orders grow. The result allows arbitrary local module types, including noninvertible modules, but does not claim equidistribution for an individual orbit or cover composite degrees.
Prime Predecessor Factors Converge to the Poisson–Dirichlet Law
An OpenAI preprint proves a conjecture about the prime factors of \(p-1\): for a uniformly sampled large prime \(p\), any fixed number of the largest prime-factor occurrences, measured by their shares of \(\log(p-1)\), converge jointly to the leading pieces of a Poisson–Dirichlet partition with parameter one. The result is a limit in distribution, not a claim that each individual prime has a typical factorization. Its proof combines estimates for factor counts in arithmetic progressions with a size-biased sampling argument that yields the stick-breaking law.
A Uniform Finiteness Theorem for Elliptic-Square Points on Curves in A₂
An OpenAI preprint claims that a curve in the moduli space of principally polarized abelian surfaces contains only finitely many points isogenous to the square of a non-CM elliptic curve, even when the elliptic curve and isogeny vary. The result applies to closed, irreducible curves defined over the algebraic numbers that are Hodge generic, with no boundary condition. In the proof strategy, comparisons at two auxiliary prime levels constrain the targets; height and orbit-size estimates then turn those local constraints into finiteness.
A 2-Converse Yields Goldfeld’s Half-and-Half Density for Elliptic-Curve Twists
A September 2026 manuscript claims to prove Goldfeld’s density prediction for every elliptic curve over the rationals: among its quadratic twists, half have analytic rank zero and half rank one. Its key step is a converse linking arithmetic information to analytic rank: when a twist’s full two-power Selmer group has corank zero or one, the authors claim its analytic rank matches that corank. They then use a distribution theorem for Selmer coranks to derive the predicted densities.
Normal Subgroups Over Global Function Fields Are Determined by Anisotropic Places
An October 2026 manuscript claims that the Margulis–Platonov conjecture holds for absolutely almost simple, simply connected groups over global function fields, including fields of characteristic two. It argues that every normal subgroup not contained in the center is determined by its image at the finitely many places where the group is anisotropic: specifically, it is the inverse image of an open normal subgroup of the product of the corresponding local groups. The proof reduces the claim to showing that a finite gap between a subgroup generated by powers and its local closure must vanish.
A Single Exterior Sheet Can Coexist With Disk Degrees Divisible by p
An OpenAI preprint, *Étale covers with a prescribed exterior sheet*, constructs a finite étale cover of a \(p\)-adic curve that is trivial on one sheet over the exterior of fixed disks, while every component above each disk has degree divisible by a chosen prime \(p\). The authors’ argument first builds cycles above the disks, then uses a group-theoretic quotient to enforce the degree condition on every disk component without losing the exterior sheet.
Every Fixed Addition Table That Covers Large Primes Also Produces Infinitely Many Composites
A September 2026 manuscript, *The Additive Indecomposability of the Primes*, argues that no two fixed sets of nonnegative integers, each containing at least two elements, can have all their pairwise sums match the primes apart from finitely many exceptions. The authors’ claim is stronger than ruling out an exact representation: any such addition table must produce infinitely many composites or miss infinitely many primes. Their proof combines constraints on residues modulo primes with estimates linking the summands’ arithmetic structure to the requirement that every sufficiently large prime be represented.
Unipotent Monodromy Preserves Nonzero Hecke Eigensheaves in Positive Characteristic
The preprint *Constructible tame Hecke eigensheaves in positive characteristic* proves an existence result for nonzero, constructible Hecke eigensheaves associated with Zariski-dense local systems and regular unipotent tame monodromy at a puncture. Its construction specializes first from characteristic zero and then to positive characteristic, under stated hypotheses on the curve, group and characteristic. To address the risk that removing auxiliary flag data erases cohomology, the article makes explicit a local coinvariant argument for nonvanishing; it does not develop the broader globalization of the trace identity behind that argument.
Non-Central Normal Subgroups Are Exact Preimages of Local Subgroups
OpenAI’s September 2026 manuscript on the Margulis–Platonov conjecture claims that, for absolutely almost simple, simply connected groups over number fields, every non-central normal subgroup of the rational points is the exact preimage of an open normal subgroup in a finite product of compact local groups. The result does not require the normal subgroup to be closed or finitely generated. The proof reduces the classification to a finite gap between a subgroup and the closure of its local image, then rules out that gap through arithmetic and finite-group arguments.
Perplexity Automations Carry Recurring Work Forward With Run History
Perplexity Academy’s guide to Computer Automations argues that recurring work can be handled as a persistent job rather than restarted from scratch each time. Users define the outcome, connected tools and review points, then set the job to run on a schedule or when an event occurs in a supported app; Computer uses the assignment and its history to carry the work forward. The guide also stresses that permissions govern what Computer can do, while credits are used when it performs the work—not while it waits for a trigger.
Computer Analytics Links Daily Usage Oversight to Enterprise Reporting
Perplexity’s Computer Analytics separates day-to-day oversight from reporting and planning: administrators can use a daily refreshed dashboard to track organization-wide credit use, top users, connectors, skills, projects, artifacts, and task duration. The company says enterprise customers can use its organization-scoped Analytics API to pull the same usage data into business-intelligence or finance systems, while individual users retain access only to their own activity.
Search Handles Answers; Computer Executes Connected Work
Perplexity Academy draws a practical line between its Search and Computer tools: Search is for questions where a cited answer is the finished product, while Computer is for work that must move across systems, produce an artifact, or run repeatedly. The company says Search can handle complex analysis—from document summaries to valuations—so long as the result is bounded and verifiable; Computer is intended to connect data and tools, build deliverables such as dashboards or reports, and send scheduled updates.
Perplexity Brings Computer Agent Workflows Into the Mac Desktop
Perplexity Academy presents Personal Computer as a Mac-based extension of Perplexity Computer, designed to let the system act on the files, apps, folders, browser tabs, and desktop context where users’ work already sits. The tutorial argues that the product’s significance is local access: users can start tasks from the Mac app, a floating desktop bar, voice mode, remote control, or Comet, while heavier processing remains on Perplexity’s servers. Its examples frame the Mac not as an endpoint for AI output, but as an operating surface for organizing folders, updating files, reviewing presentations, generating briefs, and running recurring background work.
Perplexity Computer Brings Agentic Workflows Into Microsoft Teams Threads
Perplexity’s Academy tutorial presents Computer for Microsoft Teams as an AI agent meant to run inside Teams conversations rather than in a separate Perplexity interface. The company argues that users can install Computer from the Teams marketplace, use it in direct messages for private or early-stage work, and tag it in shared channels when teammates need visibility or context. Its broader claim is that agentic workflows — research, analysis, dashboards, reports, presentations, apps and websites — can be initiated, clarified and revised in the same threads where teams already coordinate work.