Number Theory
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Paper Hilbert’s tenth problem over ℚ
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively. Manuscripts (2) October 6, 2026 ·…
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Paper The full BSD formula from low Selmer corank
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group.…
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Paper Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over \overline{\mathbb Q} with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology…
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Paper Goldfeld’s conjecture: densities and mean analytic rank
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist…
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Paper Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G{\mathbb Q} unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves…
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Paper The local p-adic section conjecture and global consequences
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale…
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Paper The Margulis–Platonov conjecture over global fields
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract…
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Paper Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan
Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over \overline{\mathbb F}q under the four stated Lie-theoretic characteristic hypotheses.…
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Paper The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including \zeta(s), is zero-free in \Re s\gt 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over…
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Paper Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K^{\mathrm M}1/\ell, K^{\mathrm M}2/\ell, and their product. These data recover the perfect closure and…
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Paper Torus-packet equidistribution in prime, quartic, and sextic degrees
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order…
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Paper Uchida’s conjecture for open homomorphisms of Galois groups
Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the…
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Paper Primitive roots for every admissible integer base
Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least ca x/(\log x)^2 primes in every sufficiently large interval…
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Paper Modularity of elliptic curves over imaginary quadratic fields
Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place. Manuscript…
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Paper Uniformly bounded components of Gaussian-prime graphs
Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D…
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Paper Potential integral density on curve character varieties
Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr…
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Paper Positive lower density of large prime gaps
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed C\log pn, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices…
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Paper A quadratic bound for Jacobsthal’s function
Answers Jacobsthal's quadratic-bound question: every interval of Ck^2 consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute…
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Paper Patterson's first moment for cubic Gauss sums
Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit…
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Paper The weak inhomogeneous Duffin–Schaeffer conjecture
Proves that for every real shift γ and finite-valued $`\psi:\mathbb N\to September 25, 2026 · · cite as `` We prove the weak inhomogeneous…
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Paper An asymptotic formula for the number of totients
Gives an asymptotic equivalent for the number V(x) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular,…
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Paper Short Egyptian fractions
Every rational a/b with 1\le a\lt b is a sum of O(\log\log b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian…
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Paper Ordinary two-point correlations and the corrected Elliott conjecture
Proves the ordinary two-point Chowla conjecture, with a bound O(X/(\log X)^c) for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary…
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Paper Squarefree quartics and power-free polynomial values
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally,…
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Paper Zilber–Pink in abelian varieties and the Siegel threefold
Proves the abelian Zilber–Pink conjecture over \overline{\mathbb Q}: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the…
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Paper Prime-factor statistics of p-1
Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law \mathrm{PD}(1) as p ranges uniformly over primes up to x and…
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Paper Independent largest prime factors of consecutive integers
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1 are asymptotically independent in ordinary natural density. In particular, the integers…
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Paper Ostmann’s inverse Goldbach conjecture
Proves that no finite modification of the primes can be written as A+B with A,B\subseteq\mathbb Z{\ge0} each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive…
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Paper The irrationality exponent of \pi is 2
Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/q satisfies |\pi-p/q|\ge q^{-2-\varepsilon}. This also proves convergence…
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Paper Irrationality of Catalan’s constant
Proves that Catalan's constant G=\sum{j\ge0}(-1)^j/(2j+1)^2 is irrational. () Manuscript September 24, 2026 · · cite as `\cite{OAI:Catalans-constant-is-irrational-September-24-2026}` We prove that Catalan's constant…
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