Landau's problems refer to a list of open problems in physics and mathematics that were posed by the renowned Soviet physicist Lev Landau. These problems primarily focus on theoretical issues in condensed matter physics, statistical mechanics, and other areas where Landau made significant contributions. One of the most famous of these problems is related to the nature of phase transitions in materials and the theoretical understanding of critical phenomena.
The Kummer–Vandiver conjecture is a statement in number theory concerning the behavior of cyclotomic fields and the behavior of the class groups associated with certain algebraic number fields. More specifically, it deals with the properties of the class number of the field obtained by adjoining a primitive \( n \)-th root of unity to the rationals.
Hilbert's ninth problem, one of the famous problems posed by the mathematician David Hilbert in 1900, focuses on the topic of mathematical logic and the foundations of mathematics. More specifically, it asks about the axiomatizability of physics, particularly the question of whether the axioms of physics can be formulated in a way that is both complete and consistent, using a finite set of axioms or a set of axioms that can be finitely derived.
Hermite's problem, named after the French mathematician Charles Hermite, refers to an important question in the theory of numbers that concerns the representation of numbers as sums of squares. Specifically, the problem seeks to establish conditions under which a natural number can be expressed as a sum of squares of integers. One of the notable results related to Hermite's problem is a theorem concerning the number of ways a given positive integer can be expressed as a sum of two squares.
Hall's conjecture is a concept in combinatorics and graph theory, specifically related to the properties of perfect matchings in bipartite graphs. The conjecture states that a certain condition involving the size of subsets of one partition of a bipartite graph must hold for the graph to contain a perfect matching.
The Grothendieck–Katz \( p \)-curvature conjecture is a conjecture in the field of algebraic geometry and number theory, particularly dealing with \( p \)-adic differential equations and their connections to the geometry of algebraic varieties. The conjecture is concerned with the behavior of differential equations over fields of characteristic \( p \), especially in relation to \( p \)-adic representations and the concept of \( p \)-curvature.
Grimm's conjecture is a mathematical hypothesis in number theory concerning prime numbers. It specifically deals with the distribution of prime gaps and the existence of infinitely many prime pairs. Proposed by mathematician Ronald Graham, it asserts that for every integer \( n \), there are infinitely many pairs of prime numbers \( (p, p + n) \) such that both \( p \) and \( p + n \) are prime.
Greenberg's conjecture is a statement in the field of number theory related to the study of Galois representations and p-adic fields. Specifically, it deals with the relation between the arithmetic of cyclotomic fields and the behavior of certain types of Galois representations.
Goldbach's conjecture is a famous unsolved problem in number theory, proposed by the Prussian mathematician Christian Goldbach in 1742. The conjecture posits that every even integer greater than two can be expressed as the sum of two prime numbers. For example, the number 4 can be expressed as \(2 + 2\), 6 can be expressed as \(3 + 3\), and 8 can be expressed as \(3 + 5\).
Gillies' conjecture is a hypothesis in the field of number theory that relates to the distribution of powers of prime numbers. Specifically, it suggests that if you take any finite set of integers and consider their product, the resulting product is often composite. The conjecture posits that a certain rational expression, derived from the powers of prime numbers that comprise the integers in the set, will eventually yield a non-zero value under specific conditions.
Gilbreath's conjecture is an observation in number theory regarding the differences between consecutive prime numbers. It asserts that if you take the sequence of prime numbers and repeatedly form new sequences by subtracting each prime from the next one, the resulting sequences will always contain primes. More formally, consider a list of prime numbers \( p_1, p_2, p_3, \ldots \).
The Four Exponentials Conjecture is a mathematical conjecture that relates to the asymptotic behavior of certain types of differential equations, specifically those that involve exponential growth. More formally, the conjecture is focused on the minimal growth rates of solutions to systems of certain differential equations.
Firoozbakht's conjecture is a mathematical conjecture related to the properties of prime numbers, specifically concerning the gaps between consecutive prime numbers.
A Fibonacci prime is a Fibonacci number that is also a prime number. The Fibonacci sequence is defined recursively, starting with the numbers 0 and 1, and each subsequent number is the sum of the two preceding ones.
The Feit-Thompson conjecture is a statement in group theory, which is a branch of mathematics that studies the algebraic structure known as groups. The conjecture was proposed by Walter Feit and John G. Thompson in their famous work in the 1960s on finite groups. The conjecture itself states that every finite group of odd order is solvable.
Euler's constant, commonly denoted by the symbol \( e \), is a mathematical constant that is approximately equal to 2.71828. It serves as the base of the natural logarithm and is extensively used in various areas of mathematics, particularly in calculus, complex analysis, and number theory.
The Erdős–Turán conjecture on additive bases is a famous conjecture in additive number theory, which is concerned with the representation of integers as sums of elements from specific sets, known as additive bases. Formally, the conjecture can be stated as follows: Let \( B \) be a set of integers.
The Elliott-Halberstam conjecture is a significant hypothesis in number theory, specifically in the field of analytic number theory, dealing with the distribution of prime numbers in arithmetic progressions. It was formulated by the mathematicians Paul Elliott and Harold Halberstam in the 1960s. The conjecture asserts that there is a specific form of "density" of primes in arithmetic progressions that can be used to improve results concerning the distribution of primes.
Dickson's conjecture is a hypothesis in number theory proposed by the mathematician Leonard Eugene Dickson in 1904. It relates to the distribution of prime numbers and specifically addresses the behavior of prime numbers in arithmetic progressions. The conjecture states that for any given set of integer numbers \(a_1, a_2, ...
Cramér's conjecture is a hypothesis in number theory related to the distribution of prime numbers. It was proposed by the Swedish mathematician Harald Cramér in 1936. The conjecture specifically addresses the gaps between consecutive prime numbers. Cramér's conjecture suggests that the gaps between successive primes \( p_n \) and \( p_{n+1} \) are relatively small compared to the size of the primes themselves.