Polignac's conjecture
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| Polignac's conjecture | |
|---|---|
| Name | Polignac's conjecture |
| Field | Number theory |
| Proposer | Alphonse de Polignac |
| Year | 1849 |
| Status | Open (conjectured) |
Polignac's conjecture is a conjecture in analytic number theory proposing that every even integer occurs infinitely often as the difference between consecutive prime numbers. It was formulated in the nineteenth century and remains unresolved despite significant progress by researchers associated with sieve methods, probabilistic models, and computational projects. The conjecture connects to major topics and figures in mathematics such as Goldbach's conjecture, Twin prime conjecture, Sieve theory, Hardy–Littlewood conjectures, and advances by scholars from institutions like École Polytechnique, University of Cambridge, and Princeton University.
Statement of the conjecture
Polignac proposed that for every even positive integer 2k there exist infinitely many pairs of consecutive primes p_n and p_{n+1} with p_{n+1} − p_n = 2k, linking his claim to the study of prime gaps investigated by Joseph-Louis Lagrange, Adrien-Marie Legendre, Chebyshev, Bernhard Riemann, and later by G. H. Hardy and J. E. Littlewood. The statement implies the special cases of the Twin prime conjecture (2k = 2) and relates to the distribution questions addressed by Atle Selberg, Enrico Bombieri, and Alan Baker.
Historical background and origin
Alphonse de Polignac formulated the conjecture in correspondence and publications in 1849 during a period when figures like Carl Friedrich Gauss, Adrien-Marie Legendre, and Peter Gustav Lejeune Dirichlet were developing prime distribution ideas. The conjecture was situated amid nineteenth-century efforts that culminated in the Prime Number Theorem proven by Jacques Hadamard and Charles-Jean de la Vallée Poussin, and it influenced twentieth-century inquiries by Paul Erdős, Viktor Kac, and members of the Bourbaki group. Subsequent attention came from researchers at Université Paris-Sud, University of Oxford, and Harvard University who linked Polignac’s idea to emerging sieve and analytic techniques.
Partial results and related theorems
Significant partial progress includes bounded gap results by Yitang Zhang, who proved the existence of infinitely many prime pairs within some fixed bound, and subsequent improvements by the collaborative Polymath Project, driven by participants from Massachusetts Institute of Technology, Cambridge University, and University of Toronto. The Maynard–Tao approach developed by James Maynard and Terence Tao extended sieve ideas pioneered by Atle Selberg and refined methods related to the Bombieri–Vinogradov theorem and Elliott–Halberstam conjecture. Work by Goldston, Pintz, and Yıldırım connected small gaps to correlations studied by Montgomery and Soundararajan, while conditional results depend on hypotheses from Riemann Hypothesis-adjacent research by Bernhard Riemann and Alan Turing.
Heuristic arguments and probabilistic models
Heuristic support for Polignac’s conjecture stems from the Hardy–Littlewood prime k-tuples conjecture, probabilistic models developed by Cramér, and random sequence analogies investigated by Paul Erdős and Mark Kac. Models using sieve weights and correlation heuristics echo constructions by G. H. Hardy and J. E. Littlewood and predictive formulas reminiscent of the Prime Number Theorem density estimates associated with Adrien-Marie Legendre and Carl Friedrich Gauss. Probabilistic number theory techniques from researchers at Institute for Advanced Study and Clay Mathematics Institute inform expected asymptotics for occurrences of fixed even gaps.
Generalizations and variants
Generalizations include conjectures about k-tuples of primes encapsulated by the Hardy–Littlewood k-tuple conjecture, extensions to arithmetic progressions inspired by Dirichlet's theorem on arithmetic progressions, and formulations over algebraic number fields influenced by Emil Artin and Helmut Hasse. Variants consider prime gaps in analogues like function field settings studied by André Weil and Alexander Grothendieck, and density questions linked to conjectures by Vinogradov and I. M. Vinogradov concerning additive representations related to Goldbach's weak conjecture.
Computational evidence and numerical verification
Extensive computations by projects at University of Tennessee, University of Minnesota, Google, and collaborative efforts like the PrimeGrid project have cataloged large prime gaps and many instances of even gaps up to substantial bounds, complementing historical tabulations by D. J. Bernstein and compilations by Nicely, Thomas related to the Mersenne primes and Cunningham project. Empirical data up to very high numeric ranges show frequent occurrences of small even gaps consistent with Polignac’s prediction, aligning with expectations from the Hardy–Littlewood constants and statistical models used by researchers at Los Alamos National Laboratory and Bell Labs.