List of unsolved mathematical problems

 

Certainly! Here's a list of some well-known unsolved problems in mathematics:

1. Riemann Hypothesis: Concerns the distribution of the zeros of the Riemann zeta function. It states that all non-trivial zeros of the zeta function have a real part equal to 1/2.

2. Goldbach's Conjecture: Posits that every even integer greater than 2 can be expressed as the sum of two primes.

3. Birch and Swinnerton-Dyer Conjecture: A conjecture in the field of number theory that relates the number of rational points on an elliptic curve to the behavior of its L-function at a certain point.

4. Collatz Conjecture (3n + 1 Problem): This conjecture concerns a simple iterative process applied to integers, which states that starting from any positive integer, if it is even, divide it by 2, and if it's odd, multiply it by 3 and add 1, and repeat, eventually the sequence will reach 1.

5. Navier-Stokes Existence and Smoothness: The problem of existence and smoothness of solutions to the Navier-Stokes equations, which describe the motion of viscous fluid substances.

6. Yang-Mills Existence and Mass Gap: This problem involves proving the existence of Yang-Mills theory solutions and the presence of a mass gap in quantum Yang-Mills theory.

7. P versus NP Problem: Asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.

8. Hodge Conjecture: Concerns the cohomology classes of complex algebraic varieties and their relationship with rational algebraic cycles.

9. Bounded Gaps Conjecture: Suggests that there are infinitely many pairs of prime numbers that differ by a finite number (bounded gap).

10. Erdős Discrepancy Problem: Asks whether for any infinite sequence of +1 and -1, there always exists a subsequence whose partial sums have unbounded absolute values.

These are just a few examples, and there are many more fascinating unsolved problems across various branches of mathematics.

Certainly! Here are a few more unsolved problems in mathematics:

11. Birch-Swinnerton-Dyer Conjecture: It deals with elliptic curves and their rational points. It conjectures a connection between the rank of the group of rational points on an elliptic curve and the behavior of its associated L-function at a certain critical point.

12. The Twin Prime Conjecture: Posits that there are infinitely many pairs of prime numbers that have a difference of 2 (e.g., 3 and 5, 11 and 13, 17 and 19, etc.).

13. Beal's Conjecture: Proposed by Andrew Beal in 1993, it states that if $A^x+B^y=C^z$, where $A$, $B$, $C$, $x$, $y$, and $z$ are positive integers and $x$, $y$, and $z$ are all greater than 2, then $A$, $B$, and $C$ must have a common prime factor.

14. The Continuum Hypothesis: Proposed by Georg Cantor, it asks whether there is a set whose cardinality is strictly between that of the integers and the real numbers. In other words, it asks whether there exists a set with cardinality equal to the cardinality of the real numbers but strictly greater than the cardinality of the integers.

15. The Hairy Ball Theorem: This theorem states that there is no way to continuously comb a hairy ball flat without creating a cowlick, assuming that each hair is a continuous curve on the surface of the ball. The problem is unsolved in higher dimensions.

16. The Hadwiger-Nelson Problem: It asks for the minimum number of colors needed to color the plane such that no two points at distance 1 from each other have the same color.

17. The Moving Sofa Problem: This geometric problem asks for the shape of the largest sofa that can be moved around a 90-degree corner in a hallway of unit width.

18. The Erdős-Straus Conjecture: It conjectures that for every integer $n\ge 2$, there exist positive integers $a$, $b$, and $c$ such that $\frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.

These are just a few more examples of unsolved problems in mathematics, each representing unique challenges and areas of research within the field.

Certainly! Here are a few more unsolved problems in mathematics:

19. The Moving Sofa Problem: This geometric problem asks for the shape of the largest sofa that can be moved around a 90-degree corner in a hallway of unit width.

20. The Four Color Theorem: This theorem states that any map on a plane can be colored using at most four colors in such a way that no two adjacent regions have the same color

Certainly! Here are a few more unsolved problems in mathematics:

29. The $n$-Body Problem: This problem, originating in classical mechanics, seeks to determine the motion of $n$ point masses under the influence of their mutual gravitational or electromagnetic interactions. While solutions exist for certain special cases (such as the two-body problem), a general solution for $n\ge 3$ remains elusive.

30. The $P=BPP$ Problem: This problem lies within the field of computational complexity theory and asks whether the complexity classes $P$ (deterministic polynomial time) and $BPP$ (probabilistic polynomial time with bounded error) are equal.

31. The Subset Sum Problem: Given a set of integers, the problem is to determine whether there exists a subset that sums to zero (or any given target sum). This problem is known to be NP-complete, meaning that no efficient algorithm exists to solve all instances of it.

32. The Collatz Conjecture Generalization: Extending the original Collatz Conjecture, variations seek to explore similar sequences generated by different rules or modifications to the original problem, such as changing the multiplication factor or considering different starting conditions.

33. The $n$-Queens Problem: In the realm of combinatorial optimization and chessboard configurations, this problem asks for the placement of $n$ queens on an $n\times n$ chessboard such that no two queens threaten each other. While solutions exist for certain values of $n$, a general algorithm to solve the problem for any $n$ remains an open challenge.

34. The Modular Inverse Problem: Given an integer $a$ and a modulus $m$, the problem is to find an integer $x$ such that $ax\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}m$. While efficient algorithms exist for certain cases (e.g., when $m$ is prime), finding a general algorithm for all cases remains an open problem.

35. The Busy Beaver Problem: In computability theory, this problem seeks to find the maximum number of steps a Turing machine with a given number of states can perform before halting when started on a blank tape. While solutions are known for machines with a small number of states, finding the Busy Beaver numbers for larger numbers of states is a challenging open problem.

These are just a few more examples of unsolved problems in mathematics, spanning various branches of the field and highlighting the breadth of ongoing research and exploration in mathematics.

Of course! Here are a few more unsolved problems in mathematics:

36. The Ruziewicz Problem: This problem in functional analysis asks whether every bounded linear operator on a Banach space can be approximated by finite-rank operators in an appropriate sense.

37. The Mertens Conjecture: This conjecture concerns the behavior of the Mertens function, which is defined as the sum of the Moebius function up to a given point. It states that the absolute value of the Mertens function is always less than or equal to the square root of the given point.

38. The Artin Conjecture on Primitive Roots: This conjecture, proposed by Emil Artin, concerns the existence of primitive roots modulo a prime number. It suggests that for any prime number $p$, there are infinitely many primitive roots modulo $p$.

39. The Perfect Cuboid Problem: This problem in geometry asks whether there exists a cuboid with integer side lengths, other than a cube, whose face diagonals and space diagonals are all rational numbers.

40. The Eisenstein Integer Problem: This problem involves determining whether there exist three distinct Eisenstein integers (complex numbers of the form $a+b\omega$, where $a$, $b$ are integers and $\omega$ is a primitive cube root of unity) whose sum is zero.

41. The Proofs of Fermat's Last Theorem: While Fermat's Last Theorem itself has been proved by Andrew Wiles in 1994, there remain open questions related to alternative or simplified proofs, as well as extensions of the theorem to higher powers.

42. The Polignac's Conjecture: This conjecture, similar to the Twin Prime Conjecture, suggests that there are infinitely many pairs of consecutive odd numbers $p$ and $p+2$ that are both prime.

43. The Ramsey Theory: Ramsey theory involves studying the emergence of order within chaotic structures. Many specific conjectures within Ramsey theory remain unsolved, such as the van der Waerden conjecture and the Erdős-Hajnal conjecture.

44. The Continuation of the Collatz Conjecture: Beyond the classic Collatz conjecture, various extensions and generalizations of the problem exist, each posing unique challenges and remaining open questions.

These are just a few more examples of intriguing unsolved problems in mathematics, showcasing the depth and diversity of ongoing mathematical research.

Certainly! Here are a few more unsolved problems in mathematics:

45. The Conway's Thrackle Conjecture: This problem, proposed by John Conway, concerns the study of thrackles - configurations of points and curves with certain properties. The conjecture states that for any finite set of points in the plane, there exists a thrackle with the same set of points as vertices.

46. The Smale Conjecture: Proposed by Stephen Smale, this conjecture asserts that for any closed, simply connected, smooth manifold of dimension greater than four, the number of Morse critical points of a generic smooth function on the manifold is bounded below by a linear function of the Morse index.

47. The Hadamard's Maximal Determinant Problem: This problem asks for the maximum determinant that a real or complex square matrix of a given size can have. While determinants of specific sizes have been calculated, finding a general formula or algorithm to determine the maximal determinant for arbitrary sizes remains an open problem.

48. The Sum-Free Sets Problem: In additive combinatorics, this problem asks whether there exists an integer $N$ such that any set of $N$ integers contains three distinct elements whose sum is not divisible by 3.

49. The Word Problem for Groups: This problem involves determining whether there exists an algorithm that can decide, given a finitely presented group and a word in the generators of the group, whether the word represents the identity element of the group.

50. The Schanuel's Conjecture: This conjecture in transcendental number theory asserts a quantitative relationship between transcendental numbers and linearly independent algebraic numbers.

51. The Quantum Unique Ergodicity Conjecture: In quantum chaos, this conjecture suggests that generic quantum systems whose classical counterparts are ergodic also exhibit a unique ergodic behavior at the quantum level.

52. The Kepler Conjecture: This problem, named after Johannes Kepler, asks for the optimal arrangement of spheres in three-dimensional space to minimize the density of the packing. The conjecture has been proven by Thomas Hales with extensive computer assistance.

53. The Discrete Logarithm Problem: This problem, particularly in the context of cryptographic protocols, involves finding an exponent $x$ such that $a^x\equiv b\mathrm{m}\mathrm{o}\mathrm{d}p$ for given $a$, $b$, and $p$, where $p$ is a prime number.

These are just a few more examples of fascinating unsolved problems in mathematics, each representing unique challenges and avenues for exploration within the field.

Certainly! Here are a few more unsolved problems in mathematics:

54. The Existence of Odd Perfect Numbers: A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). All known perfect numbers are even, and it's unknown whether odd perfect numbers exist.

55. The Distribution of Primes: While there are several conjectures and results related to the distribution of prime numbers, such as the Prime Number Theorem, there are still many open questions regarding the finer details of how primes are distributed among the integers.

56. The Determination of the Decimal Expansion of Pi and e: While the decimal expansions of the mathematical constants $\pi$ (pi) and $e$ (Euler's number) are known to be non-repeating and non-terminating, it's an open problem to determine specific properties of their digits, such as the distribution of digit frequencies.

57. The Classification of Finite Simple Groups: This monumental project, known as the Classification Theorem for Finite Simple Groups, was completed in 2004. However, some gaps and exceptional cases remain to be fully resolved, particularly related to the classification of sporadic groups.

58. The Generalized Fermat Equation: The equation $a^n+b^n=c^n$ is known as the Fermat equation, famously conjectured by Pierre de Fermat in the 17th century. While Fermat's Last Theorem addresses the case when $n$ is a positive integer greater than 2, the general equation for arbitrary $n$ remains unsolved.

59. The Classification of Singularities in Algebraic Geometry: Singularities arise in algebraic varieties and play a crucial role in understanding their geometric properties. While significant progress has been made, a complete classification of singularities in arbitrary dimensions and contexts remains an open problem.

60. The Existence of Harmonic Primes: A harmonic prime is a prime number whose reciprocal has a repeating decimal expansion with period length equal to the prime itself. It is not known whether there are infinitely many harmonic primes.

61. The Rigidity of Three-Dimensional Manifolds: This problem concerns whether the geometry of a three-dimensional manifold can determine its topology uniquely. Various conjectures and results in geometric topology relate to this broader question.

62. The Existence of Infinitely Many Twin Primes in Arithmetic Progression: While it's known that there are infinitely many twin primes (pairs of primes that differ by 2), it's an open problem whether there are infinitely many such pairs that also form part of an arithmetic progression.

These are just a few more examples of unsolved problems in mathematics, each representing significant challenges and areas of ongoing research in the field.

Certainly! Here are a few more unsolved problems in mathematics:

63. The Periodicity of the Collatz Function: While the Collatz conjecture (also known as the 3x+1 problem) asserts that every positive integer eventually reaches 1 when subjected to the Collatz function, the question of whether there exist cycles or periodic behavior for certain values remains unresolved.

64. The Existence of Quantum Error-Correcting Codes of Optimal Length and Distance: In quantum information theory, the search for quantum error-correcting codes with optimal parameters (length and distance) remains a challenging open problem with applications in quantum computing and communication.

65. The Frobenius Coin Problem: Given a set of coin denominations, the Frobenius coin problem asks for the largest integer amount of money that cannot be obtained by using a finite number of coins from the set. While solutions exist for certain cases, finding a general formula or algorithm for all cases remains elusive.

66. The Waring Problem: Named after Edward Waring, this problem asks whether every positive integer can be expressed as the sum of a fixed number of $k$-th powers of positive integers, where $k$ is a fixed positive integer greater than 1.

67. The Unique Games Conjecture: This conjecture, in the field of approximation algorithms and computational complexity theory, asserts that a certain problem, known as the Unique Games problem, is hard to approximate within a certain factor. Its resolution would have significant implications for computational hardness.

68. The Zariski Cancellation Problem: This problem in algebraic geometry concerns whether the ring of polynomial functions on an affine variety determines the variety uniquely up to isomorphism.

69. The Hilbert-Smith Conjecture: Proposed by David Hilbert and Poincaré, this conjecture in topology states that any topological group acting continuously and effectively on a manifold must have a fixed point.

70. The Traveling Salesman Problem with Neighborhoods: This variant of the classic Traveling Salesman Problem involves finding the shortest tour that visits a set of neighborhoods (rather than individual points) at least once, returning to the starting point after visiting each neighborhood.

71. The Decidability of the Word Problem for Finitely Presented Semigroups: In semigroup theory, this problem asks whether there exists an algorithm to determine whether a given word in the generators of a finitely presented semigroup represents the identity element.

72. The Inverse Galois Problem for Finite Groups: This problem, a variation of the Inverse Galois Problem, asks whether every finite group arises as the Galois group of some Galois extension of the rational numbers.

These are just a few more examples of intriguing unsolved problems in mathematics, highlighting the diversity and depth of ongoing research in the field.

Certainly! Here are a few more unsolved problems in mathematics:

73. The Polynomial Identity Testing Problem: This problem involves determining whether a given multivariate polynomial with integer coefficients is identically zero. The challenge is to find efficient algorithms for this problem, especially in cases where the number of variables and the degree of the polynomial are large.

74. The Existence of Continuous Nowhere-Differentiable Functions: It is known that there exist continuous functions that are nowhere differentiable, such as the famous Weierstrass function. However, it remains an open problem whether there exist continuous functions that are nowhere locally bounded.

75. The Existence of Transcendental Numbers in Specific Sets: While it is known that transcendental numbers exist, it remains an open question whether specific sets of real numbers (such as the set of real numbers that can be defined by polynomials with integer coefficients) contain transcendental numbers.

76. The Existence of Integer Solutions to Diophantine Equations: Diophantine equations are polynomial equations with integer coefficients, and the problem of determining whether they have integer solutions is a central topic in number theory. Many specific Diophantine equations, such as the generalized Fermat equation, remain unsolved.

77. The Existence of Graphs with Specific Ramsey Numbers: Ramsey theory deals with the emergence of order in random structures. Determining specific Ramsey numbers, which represent the size of graphs required to guarantee certain properties, remains an open problem for many cases.

78. The Existence of a Polynomial-Time Algorithm for Integer Factorization: While integer factorization is known to be in NP (the class of decision problems that can be verified in polynomial time), it remains an open question whether there exists a polynomial-time algorithm for this problem.

79. The Existence of Continuous Surjections from the Real Line to the Plane: It is known that there exist continuous surjections from the plane to the real line, but it remains an open problem whether the reverse is true.

80. The Existence of Perfect Cubes that are Sums of Consecutive Positive Integers: It is known that there exist perfect squares that are sums of consecutive positive integers (e.g., $1^2+2^2=3^2$), but it remains an open problem whether perfect cubes with similar properties exist.

81. The Existence of Quasirandom Graphs with Specific Properties: Quasirandom graphs are graphs that exhibit properties similar to random graphs. Determining the existence of quasirandom graphs with specific properties remains an open problem in graph theory.

82. The Existence of Unramified Regular Coverings of Certain Surfaces: It is known that every compact Riemann surface admits unramified regular coverings, but determining the existence of such coverings with specific properties remains an open problem.

These are just a few more examples of unsolved problems in mathematics, each representing unique challenges and areas of ongoing research in the field.