Top 50 Most Unsolved and Hardest Math Equations in the World

1. Riemann Hypothesis

• Equation: All non-trivial zeros of the Riemann zeta function have real part 1/2.
• ζ(s) = ∑ (1/n^s), for n = 1 to ∞
• Founder: Bernhard Riemann (1859)
• Prize: $1,000,000 (Clay Millennium Prize)
• Condition: Prove or disprove that all non-trivial zeros lie on Re(s) = 1/2.
• Status: Unsolved
• Field: Number Theory, Complex Analysis

2. Birch and Swinnerton-Dyer Conjecture

• Equation: Rank of an elliptic curve = order of zero of its L-function at s = 1.
• L(E, s) = analytic function associated with elliptic curves
• Founder: Bryan Birch & Peter Swinnerton-Dyer (1960s)
• Prize: $1,000,000 (Clay Prize)
• Condition: Provide proof connecting the rank of an elliptic curve with its L-function behavior at s = 1.
• Status: Unsolved
• Field: Algebraic Geometry, Number Theory

3. Navier–Stokes Existence and Smoothness

• Equation: ∂u/∂t + (u·∇)u = −∇p + ν∇²u + f, ∇·u = 0
• Founder: Claude-Louis Navier & George Gabriel Stokes (19th century)
• Prize: $1,000,000 (Clay Prize)
• Condition: Prove the existence and smoothness of 3D solutions in R³ over time.
• Status: Unsolved
• Field: Fluid Dynamics, PDEs

4. Yang–Mills Existence and Mass Gap

• Equation: Quantum Yang–Mills field equations with a positive mass gap.
• L = −(1/4)Fᵃ_μν Fᵃ^μν
• Founder: Chen-Ning Yang and Robert Mills (1954)
• Prize: $1,000,000 (Clay Prize)
• Condition: Prove that Yang–Mills theory exists and has a mass gap > 0.
• Status: Unsolved
• Field: Theoretical Physics, Quantum Field Theory

5. P vs NP Problem

• Equation: Is P = NP?
• P: Problems solvable quickly; NP: Verifiable quickly
• Founder: Stephen Cook (1971)
• Prize: $1,000,000 (Clay Prize)
• Condition: Prove or disprove that every problem whose solution can be quickly verified can also be quickly solved.
• Status: Unsolved
• Field: Computer Science, Algorithms

6. Hodge Conjecture

• Equation: Certain types of cohomology classes are algebraic cycles.
• Founder: W.V.D. Hodge (1950s)
• Prize: $1,000,000 (Clay Prize)
• Condition: Prove the link between Hodge classes and algebraic cycles.
• Status: Unsolved
• Field: Algebraic Geometry, Topology

7. Collatz Conjecture

• Equation:

• • If n is even: n → n/2

  • If n is odd: n → 3n + 1
    Repeat. Does every n reach 1?

• Founder: Lothar Collatz (1937)
• Prize: No official prize; $500 offered by Paul Erdős
• Condition: Prove that this sequence reaches 1 for all positive integers.
• Status: Unsolved
• Field: Number Theory, Dynamical Systems

8. Goldbach Conjecture

• Equation: Every even integer ≥ 4 is the sum of two prime numbers.
• Founder: Christian Goldbach (1742)
• Prize: None officially, but massive fame and prestige.
• Condition: Prove for all even integers > 2.
• Status: Unsolved
• Field: Number Theory

9. Twin Prime Conjecture

• Equation: Infinitely many primes p such that p+2 is also prime.
• Founder: Alphonse de Polignac (1849)
• Prize: $1,000,000 (offered by various sponsors like Polymath Project)
• Condition: Prove infinite number of such prime pairs.
• Status: Unsolved
• Field: Number Theory

10. Beal’s Conjecture

• Equation: If Aⁿ + Bᵐ = Cʳ, with A, B, C, n, m, r ∈ ℕ and n, m, r > 2, then A, B, C have a common factor.
• Founder: Andrew Beal (1993)
• Prize: $1,000,000 (offered by Andrew Beal)
• Condition: Prove or disprove the equation.
• Status: Unsolved
• Field: Number Theory

11. Euler’s Sum of Powers Conjecture

• Equation: a⁴ + b⁴ + c⁴ ≠ d⁴ (extended: 4th powers can’t sum to another 4th power)

• Founder: Leonhard Euler (18th century)

• Prize: None

• Condition: Prove non-existence for powers > 3.

• Status: Counterexamples found for 4th power (false)

• Field: Number Theory

12. Perfect Cuboid Problem

• Equation: Find a rectangular box where all edges, face diagonals, and the space diagonal are rational.

• Founder: Unknown

• Prize: No official prize

• Condition: Prove or disprove existence

• Status: Unsolved

• Field: Geometry, Number Theory

13. Existence of Odd Perfect Numbers

• Equation: A perfect number is one equal to the sum of its proper divisors. Are any odd?

• Founder: Euclid (ancient), ongoing study

• Prize: None officially

• Condition: Prove whether any odd perfect number exists.

• Status: Unsolved

• Field: Number Theory

14. Syracuse Problem (3x+1 again)

• Same as Collatz Conjecture

• Another name, showing its global research under multiple names.

15. The Continuum Hypothesis

• Statement: No set whose cardinality is strictly between that of integers and real numbers.

• Founder: Georg Cantor (1878)

• Prize: Solved partially (independent of ZFC axioms)

• Condition: Whether ℵ₁ = 2^ℵ₀

• Status: Independent of standard set theory (Gödel & Cohen)

• Field: Set Theory

16. Mertens Conjecture

• Equation: |M(n)| < √n, where M(n) is Mertens function

• Founder: Franz Mertens (1897)

• Status: Disproved by Odlyzko and te Riele (1985)

• Prize: None

• Field: Number Theory

17. Hadamard Conjecture

• Equation: Hadamard matrix of order 4k exists for all k

• Founder: Jacques Hadamard

• Prize: No official prize

• Status: Open for some k

• Field: Linear Algebra, Combinatorics

18. Erdős Discrepancy Problem

• Equation: Discrepancy of ±1 sequences is unbounded

• Founder: Paul Erdős

• Prize: Proved true in 2015 by Terence Tao

• Status: Solved

• Field: Combinatorics

19. Monty Hall Paradox (Solved but confusing)

• Problem: Switch or stay? What’s the best strategy?

• Founder: Named after a TV show

• Status: Solved (switching is better)

• Field: Probability

20. Hilbert’s 8th Problem

• Combines Riemann Hypothesis, Goldbach, and Twin Prime

• Founder: David Hilbert (1900)

• Prize: $1,000,000+ collectively

• Status: Largely unsolved

• Field: Number Theory

21. Bunyakovsky Conjecture

• Equation: Any irreducible integer polynomial with positive leading coefficient and no common factor in outputs takes infinitely many prime values.

• Founder: Viktor Bunyakovsky (1857)

• Prize: No official prize

• Condition: Prove the polynomial generates primes infinitely often.

• Status: Unsolved

• Field: Number Theory

22. Legendre’s Conjecture

• Equation: There is always at least one prime between n² and (n+1)².

• Founder: Adrien-Marie Legendre

• Prize: No official prize

• Condition: Prove the existence of at least one prime in every such interval.

• Status: Unsolved

• Field: Number Theory

23. Andrica’s Conjecture

• Equation: √pₙ₊₁ − √pₙ < 1 (pₙ is the nth prime)

• Founder: Dorin Andrica (1986)

• Prize: No official prize

• Condition: Prove the inequality holds for all primes.

• Status: Unsolved

• Field: Number Theory

24. Brocard’s Problem

• Equation: Find integers n, m such that n! + 1 = m²

• Founder: Henri Brocard (1876)

• Prize: No official prize

• Condition: Prove whether more than the 3 known solutions exist.

• Status: Unsolved

• Field: Number Theory

25. Catalan’s Conjecture (Solved)

• Equation: xᵃ − yᵇ = 1 ⇒ x = 3, a = 2, y = 2, b = 3 is the only solution.

• Founder: Eugène Charles Catalan (1844)

• Status: Solved by Preda Mihăilescu (2002)

• Field: Number Theory

26. Open Problem in Ramsey Theory

• Equation: R(5, 5) > ? (Exact Ramsey number unknown)

• Founder: Frank P. Ramsey

• Prize: No fixed prize

• Condition: Find the smallest number R(k, k)

• Status: Unsolved for many values

• Field: Combinatorics

27. Rota’s Conjecture

• Statement: Matroids are representable over finite fields if certain conditions hold.

• Founder: Gian-Carlo Rota

• Status: Still open for general case

• Field: Combinatorics, Matroid Theory

28. Gilbreath’s Conjecture

• Equation: First differences of consecutive primes always start with 1

• Founder: Norman Gilbreath (1958)

• Status: Verified for large primes, still unproven

• Field: Number Theory

29. Density Hales-Jewett Theorem (Generalized case)

• Statement: How many combinatorial lines exist in n-dimensional space?

• Founder: Hales & Jewett

• Condition: General density versions open

• Field: Combinatorics

30. Erdős–Straus Conjecture

• Equation: 4/n = 1/a + 1/b + 1/c for all n ≥ 2

• Founder: Paul Erdős, Ernst G. Straus

• Status: Open for n ≥ 2

• Field: Number Theory

31. Loch Ness Monster Problem

• Statement: Is there a unique noncompact surface of infinite genus with one end?

• Founder: Topologists, folklore name

• Status: Partially open

• Field: Topology

32. Rational Distance Problem

• Question: Are there three points all a rational distance from each other forming an equilateral triangle?

• Founder: Open since antiquity

• Status: No example known for non-zero side

• Field: Geometry

33. Tarski’s Circle Squaring Problem (Solved)

• Statement: Can a circle be partitioned into finitely many pieces and reassembled into a square?

• Solved: Yes, in 2019 (Grabowski, Máthé, Pikhurko)

• Field: Measure Theory

34. Rational Box Problem

• Question: Does a cube with rational side and diagonals exist?

• Status: Unsolved

• Field: Number Theory, Geometry

35. Odd Weird Numbers

• Statement: Does an odd weird number exist? (Weird: abundant but not semiperfect)

• Status: None found yet

• Field: Number Theory

36. Keller’s Conjecture in 7D

• Statement: Cube tilings in 7D always share a face

• Status: 8D counterexample found, 7D open

• Field: Geometry, Tilings

37. The Lonely Runner Conjecture

• Statement: Each runner is lonely at some time on a circular track.

• Founder: Jörg M. Wills (1967)

• Status: Open for general number of runners

• Field: Number Theory, Combinatorics

38. Odd Perfect Numbers

• Statement: Does an odd perfect number exist?

• Founder: Unknown; over 2000 years old

• Status: Still not found

• Field: Number Theory

39. Erdős–Turán Conjecture on Additive Bases

• Statement: If A is an additive basis, then A(x) must grow faster than log x.

• Status: Unsolved

• Field: Additive Number Theory

40. Noether’s Problem

• Statement: Is the field of invariants rational over base field?

• Founder: Emmy Noether

• Status: Unsolved in many cases

• Field: Algebra

41. Sun’s Conjecture

• Statement: All positive integers are sums of a square, a triangular, and a hexagonal number.

• Founder: Zhi-Wei Sun

• Status: Open

• Field: Number Theory

42. Sato–Tate Conjecture (Solved for Elliptic Curves)

• Status: Proved for elliptic curves without complex multiplication (2006)

• Open: For general motives

• Field: Algebraic Geometry

43. Schanuel’s Conjecture

• Statement: Transcendental degree of field generated by complex numbers and their exponentials is at least n

• Status: Unsolved

• Field: Transcendental Number Theory

44. Littlewood’s Conjecture

• Statement: lim inf n ||na||·||nb|| = 0 for all a, b ∈ ℝ

• Founder: J.E. Littlewood

• Status: Unsolved

• Field: Diophantine Approximation

45. Möbius Inversion in Probabilistic Number Theory

• Problem: Is the Möbius function truly random?

• Status: Ongoing research

• Field: Number Theory

46. Carmichael’s Totient Function Conjecture

• Statement: φ(n) = φ(m) ⇒ n = m is false for infinitely many n ≠ m
• Founder: Robert Carmichael
• Status: Unsolved
• Field: Number Theory

47. The Jacobian Conjecture

• Statement: Are invertible polynomial maps always invertible with a polynomial inverse?
• Founder: Ott-Heinrich Keller
• Status: Open
• Field: Algebraic Geometry

48. The Gaussian Moat Problem

• Statement: Can you walk to infinity through Gaussian primes with bounded steps?
• Status: Unsolved
• Field: Complex Analysis, Number Theory

49. Nash Blowup Problem

• Statement: Are Nash blowups smooth? (Complex algebraic varieties)
• Founder: John Nash
• Status: Unsolved
• Field: Algebraic Geometry

50. The Infinite Monkey Theorem (Formal Version)

• Statement: Can a random infinite sequence generate any text with 100% probability?
• Status: Philosophically and probabilistically complex
• Field: Probability, Information Theory

Each unsolved equation tells a unique story: some emerged from abstract thought experiments, others from trying to understand the behavior of primes, or the structure of space and time. Several of these problems connect to physics, computer science, cryptography, and even philosophy, suggesting that their solutions could unlock transformative knowledge about the universe or the fabric of logic itself.

Despite advances in computing, artificial intelligence, and collaborative research, these 50 problems continue to resist resolution. They form a symbolic mountain range in the landscape of mathematics—awaiting climbers brave enough and brilliant enough to scale them.

This comprehensive list doesn’t just name the equations—it gives you a deep look at their origin, structure, known efforts, and the massive implications their solutions might carry. Whether you’re a mathematician, researcher, educator, student, or simply curious, exploring these unsolved mathematical challenges is like peering into the edge of human understanding.

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