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+# Ten Unsolved Problems in Mathematics
+
+This document provides brief, high‑level descriptions of ten major open problems in
+contemporary mathematics.  These problems have resisted solution despite
+considerable effort and are of great theoretical importance.  They are listed
+here for educational purposes.
+
+## 1. Riemann Hypothesis
+
+The Riemann zeta function is defined for complex numbers and encodes deep
+information about the distribution of prime numbers.  Its “non‑trivial” zeros
+appear to all lie on the so‑called critical line with real part one‑half.
+Proving (or disproving) that all such zeros have real part exactly 1/2 is one
+of the Clay Mathematics Institute Millennium Prize problems.  A positive
+resolution would have far‑reaching consequences for analytic number theory and
+the error term in the Prime Number Theorem.
+
+## 2. P vs NP Problem
+
+In computer science, **P** denotes the class of problems that can be solved in
+polynomial time, while **NP** denotes the class whose solutions can be
+verified in polynomial time.  It is unknown whether these two classes are
+actually equal; a proof either way would revolutionize complexity theory and
+cryptography.  This is also a Millennium Prize problem.
+
+## 3. Navier–Stokes Existence and Smoothness
+
+The Navier–Stokes equations model the flow of incompressible fluids.  In three
+dimensions, it remains open whether smooth solutions always exist for all time
+given smooth initial data.  Establishing global regularity or proving finite
+time blow‑up is another of the Millennium Prize problems.
+
+## 4. Hodge Conjecture
+
+In algebraic geometry, the Hodge conjecture predicts which cohomology classes
+of smooth projective varieties over the complex numbers can be represented by
+algebraic subvarieties.  Proving or disproving this statement is a
+long‑standing challenge and a Millennium Prize problem.
+
+## 5. Yang–Mills Existence and Mass Gap
+
+Quantum Yang–Mills theory underlies the Standard Model of particle physics.  A
+mathematical formulation requires proving the existence of a quantum theory and
+explaining why there is a positive energy gap between the ground state and the
+first excited state (the “mass gap”).  This is another Millennium Prize
+problem.
+
+## 6. Birch and Swinnerton–Dyer Conjecture
+
+Elliptic curves are algebraic curves of great arithmetic interest.  The
+conjecture formulated by Birch and Swinnerton‑Dyer relates the rank of an
+elliptic curve (the size of its group of rational points) to the order of
+vanishing of its L‑function at s = 1.  It remains unproven and is a
+Millennium Prize problem.
+
+## 7. Goldbach’s Conjecture
+
+Proposed in the 18th century, Goldbach’s conjecture asserts that every
+even integer greater than 2 can be expressed as the sum of two prime numbers.
+Although verified computationally up to enormous bounds and supported by
+heuristic arguments, a proof or disproof is unknown.
+
+## 8. Twin Prime Conjecture
+
+Twin primes are pairs of primes (p, p + 2).  The conjecture states that there
+are infinitely many such pairs.  Despite substantial progress on bounding gaps
+between primes, the conjecture remains open.
+
+## 9. Collatz Conjecture
+
+Starting with any positive integer n, repeatedly apply the rule n -> n/2 if n
+is even and n -> 3n + 1 if n is odd.  The conjecture states that every
+starting value eventually reaches the cycle 4→2→1.  No proof exists despite
+its deceptively simple formulation.
+
+## 10. Euler–Mascheroni Constant Rationality
+
+The Euler–Mascheroni constant γ appears in analysis and number theory,
+defined as the limiting difference between the harmonic series and the natural
+logarithm.  It is unknown whether γ is rational or irrational; more
+broadly, the nature of this constant remains mysterious.