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84 lines
4.0 KiB
Markdown
84 lines
4.0 KiB
Markdown
# Unsolved Problems in Mathematics
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The problems below have resisted rigorous proofs despite significant progress.
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Each description includes a brief summary and a reference for further study.
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## 1. Riemann Hypothesis
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The hypothesis asserts that every nontrivial zero of the Riemann zeta function
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lies on the critical line with real part one-half. Resolving it would illuminate
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the distribution of prime numbers and has profound consequences throughout
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analytic number theory. [^riemann]
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## 2. P vs NP
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This problem asks whether every problem whose solution can be verified quickly
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(in polynomial time) can also be solved quickly. A proof either way would
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reshape computer science, impacting cryptography, optimisation and complexity
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theory at large. [^pvsnp]
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## 3. Birch and Swinnerton-Dyer Conjecture
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For an elliptic curve over the rationals, the conjecture links the number of
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independent rational points to the behaviour of the curve’s L-function at
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``s = 1``. It provides a deep connection between arithmetic geometry and
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analytic objects. [^bsd]
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## 4. Hodge Conjecture
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The conjecture proposes that certain cohomology classes on a non-singular
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projective variety can be represented by algebraic cycles. It remains one of
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the central open problems in algebraic geometry. [^hodge]
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## 5. Yang–Mills Existence and Mass Gap
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This problem seeks a rigorous quantum field theory for non-abelian gauge fields
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that exhibits a mass gap, meaning particles possess positive lower bounds for
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their masses. It is crucial for understanding the mathematical foundations of
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particle physics. [^yangmills]
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## 6. Navier–Stokes Existence and Smoothness
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The question asks whether smooth solutions to the three-dimensional
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Navier–Stokes equations exist for all time, or if singularities can form from
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smooth initial conditions. A resolution would clarify the mathematics of fluid
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flow. [^navierstokes]
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## 7. Goldbach’s Conjecture
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Goldbach proposed that every even integer greater than two is the sum of two
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primes. Despite overwhelming numerical evidence and partial results, a general
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proof remains unknown. [^goldbach]
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## 8. Twin Prime Conjecture
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The conjecture posits that there exist infinitely many pairs of primes separated
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by two. Recent advances have narrowed the permissible gap, but infinitude is
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still unproven. [^twinprime]
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## 9. Collatz Conjecture
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Starting from any positive integer, repeatedly applying the Collatz map (halve
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even numbers, triple odd numbers and add one) appears to eventually reach one.
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Proving this behaviour for every starting value has eluded mathematicians for
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decades. [^collatz]
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## 10. abc Conjecture
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This conjecture links the prime factors of three integers ``a``, ``b`` and
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``c`` satisfying ``a + b = c``. It predicts that, except for finitely many
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triples, ``c`` cannot be too large relative to the product of the distinct prime
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factors of ``abc``. Its resolution would unify numerous results in number
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theory. [^abc]
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[^riemann]: Clay Mathematics Institute, *The Riemann Hypothesis* (Millennium
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Prize Problem description).
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[^pvsnp]: Clay Mathematics Institute, *P vs NP Problem* (Millennium Prize
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Problem description).
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[^bsd]: Clay Mathematics Institute, *The Birch and Swinnerton-Dyer Conjecture*
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(Millennium Prize Problem description).
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[^hodge]: Clay Mathematics Institute, *The Hodge Conjecture* (Millennium Prize
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Problem description).
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[^yangmills]: Clay Mathematics Institute, *Yang–Mills and Mass Gap*
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(Millennium Prize Problem description).
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[^navierstokes]: Clay Mathematics Institute, *Navier–Stokes Equation*
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(Millennium Prize Problem description).
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[^goldbach]: Helmut Koch, *Number Theory: Algebraic Numbers and Functions*,
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Chapter 7.
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[^twinprime]: James Maynard, “Small gaps between primes,” *Annals of
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Mathematics* **181** (2015).
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[^collatz]: Jeffrey C. Lagarias, “The 3x + 1 Problem: An Annotated Bibliography
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(1963–1999),” *arXiv:math/0309224*.
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[^abc]: David Masser and Joseph Oesterlé, conference reports (1985) introducing
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the abc conjecture.
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