Unsolved problems in number theory

Unsolved problems in number theory
Name	Unsolved problems in number theory
Field	Number theory
Notable	Riemann hypothesis; Goldbach conjecture; Birch and Swinnerton-Dyer conjecture
Related	Prime number theorem; Fermat's Last Theorem; Langlands program

Unsolved problems in number theory are central questions about integers, primes, Diophantine equations, arithmetic geometry, and the analytic properties of zeta and L-functions that remain unresolved despite extensive partial results by mathematicians and institutions. These problems connect the work of historical figures such as Euclid, Eratosthenes, Pierre de Fermat, and Leonhard Euler with modern programs like the Langlands program and conjectures posed by Bernhard Riemann, G. H. Hardy, and André Weil. Many open questions stimulate research across organizations including the Clay Mathematics Institute, the Institute for Advanced Study, and universities worldwide, and relate to awards such as the Fields Medal and prizes like the Millennium Prize Problems.

Overview and classification

The landscape of unresolved questions admits several useful classifications: conjectures about primes and additive problems; Diophantine equations and rational or integral points on curves and varieties; analytic statements about zeros and value distribution of zeta and L-functions; algebraic and arithmetic geometry conjectures about motives, Galois representations, and modularity; and computational complexity and decidability problems. Major benchmark problems—e.g., the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture—anchor these classes, while conjectures like the Goldbach conjecture and the Twin Prime conjecture motivate sieve-theoretic and combinatorial investigations linked to work of Yitang Zhang, Terence Tao, and János Pintz.

Prime-related conjectures

Prime-related questions include distributional conjectures and structure statements about primes in arithmetic progressions, gaps, and patterns. The Riemann hypothesis implies strong bounds for the error term in the prime number theorem and relates to the zeros of the Riemann zeta function and general Dirichlet L-function families. The Goldbach conjecture, the Twin Prime conjecture, the Polignac conjecture, and the Prime k-tuples conjecture ask about additive decompositions and constellations of primes; progress by Vinogradov, Chen Jingrun, and later breakthroughs by Yitang Zhang and collaborators reduced gaps but left infinitude questions open. Questions about primes represented by polynomials trace to Bunyakovsky and Bateman–Horn conjecture formulations and intersect work of Heath-Brown and Green–Tao theorem on arithmetic progressions. Equidistribution in residue classes ties to Dirichlet's theorem on arithmetic progressions, Siegel zeros, and conjectural refinements like the Generalized Riemann Hypothesis.

Diophantine equations and integer solutions

Diophantine problems span existence, finiteness, and effective determination of integer or rational solutions to polynomial equations. The Mordell conjecture (proved by Gerd Faltings) and Fermat's Last Theorem (proved by Andrew Wiles) illustrate resolved major items, while the Birch and Swinnerton-Dyer conjecture predicts the rank of elliptic curves—a central unresolved statement linking heights, regulators, and L-values. The Hasse principle and its failures via counterexamples involve obstructions like the Brauer–Manin obstruction studied by Jean-Louis Colliot-Thélène. Hilbert's tenth problem for the integers (decidability) was resolved by Matiyasevich building on work by Davis–Putnam–Robinson–Matiyasevich, but analogues over fields such as the rationals or number fields remain open; related conjectures include Mazur's conjectures on rational points and uniform boundedness statements by Barry Mazur and Joseph Silverman.

Analytic problems and L-functions

Analytic number theory centers on the zeros, moments, and value distribution of zeta and L-functions. The Generalized Riemann Hypothesis (GRH) for automorphic L-functions and its implications for class groups, primality testing, and effective Chebotarev density statements remain unproven. Conjectures about subconvexity bounds, nonvanishing at special points, and the Grand Simplicity Hypothesis about linear independence of zeros are vital for equidistribution results pursued by researchers at institutions like the European Research Council-funded projects. The Sato–Tate conjecture (now proved for many cases) exemplifies distributional questions for families of L-functions; unresolved extensions to higher-dimensional motives and general automorphic representations connect to the Langlands conjectures and conjectural reciprocity laws proposed by Robert Langlands.

Algebraic and arithmetic geometry problems

In arithmetic geometry, conjectures address motives, Galois representations, and the structure of algebraic varieties over number fields. The Bloch–Kato conjecture and Beilinson conjectures predict relations between special values of L-functions and K-theory; the Tate conjecture and Hodge conjecture relate algebraic cycles to cohomology and remain central in the study of varieties over finite fields and complex numbers. The conjectural category of mixed motives and the expected properties of motivic Galois groups underpin deep questions formulated by Alexander Grothendieck and Pierre Deligne. Modularity statements generalizing Wiles's theorem to higher-dimensional varieties, and the Fontaine–Mazur conjecture about geometric Galois representations, interlink with deformation theory developed by Barry Mazur and modularity lifting techniques.

Computational and complexity questions

Computational aspects include the complexity of integer factorization, primality testing, and algorithmic decidability of Diophantine problems. While primality testing became polynomial-time via the AKS primality test and practical methods like the Elliptic Curve Primality Proving algorithm exist, the average-case and worst-case complexity of factoring is tied to cryptographic assumptions involving RSA-type schemes and quantum algorithms such as Shor's algorithm. Complexity-theoretic questions about reductions between factoring, discrete logarithm problems, and lattice problems connect to work at IBM Research, Microsoft Research, and national laboratories. Decidability over rings and fields—e.g., Hilbert's tenth problem over the rationals—remains a deep open area linking logic, model theory (as in work of Julia Robinson), and number theory.

Category:Number theory