Unsolved problems in number theory are deep questions and conjectures about integers and their properties that have not yet been resolved. Some of the most famous unsolved problems in this field include: 1. **The Riemann Hypothesis**: This conjecture concerns the distribution of the zeros of the Riemann zeta function and has profound implications for the distribution of prime numbers.
The Agoh–Giuga conjecture is a conjecture in number theory proposed by the mathematicians Yoshihiro Agoh and Giovanni Giuga. It deals with the characterization of certain prime numbers.
Andrica's conjecture is a hypothesis in number theory proposed by the Romanian mathematician Dorin Andrica in 1980. The conjecture pertains to the distribution of prime numbers and specifically conjectures a relationship between consecutive prime numbers.
Artin's conjecture on primitive roots is a conjecture in number theory proposed by Emil Artin in 1927. It concerns the existence of primitive roots modulo primes and more generally, modulo any integer.
A balanced prime is a special type of prime number that is defined in relation to its neighboring prime numbers. Specifically, a prime number \( p \) is considered to be a balanced prime if it is the average of the nearest prime numbers that are less than and greater than \( p \).
The Bateman–Horn conjecture is a hypothesis in number theory concerning the distribution of prime numbers in certain arithmetic progressions. It was proposed by the mathematicians Paul T. Bateman and Benjamin M. Horn in the 1960s. The conjecture states that for a given set of linear polynomials, the number of primes produced by these polynomials, under certain conditions, can be predicted based on the properties of the coefficients and the values at which these polynomials are evaluated.
The Bunyakovsky conjecture is a conjecture in number theory that relates to prime numbers and is named after the Russian mathematician Viktor Bunyakovsky. It deals with the existence of prime numbers generated by certain polynomial expressions.
Carmichael's totient function conjecture is a mathematical conjecture related to the properties of the Euler's totient function, denoted as \(\varphi(n)\). The conjecture is named after the mathematician Robert Carmichael. The conjecture states that for any integer \( n \) greater than \( 1 \), the inequality \[ \varphi(n) < n \] holds true, which is indeed true for all integers \( n > 1 \).
The Casas-Alvero conjecture is a statement in algebraic geometry and commutative algebra concerning the properties of certain classes of varieties, and it addresses the relationship between numerical and geometric properties of projective varieties.
The class number problem is a central question in algebraic number theory that relates to the properties of the ideal class group of a number field, specifically its class number. The class number is an important invariant that measures the failure of unique factorization in the ring of integers of a number field.
A congruent number is a natural number that is the area of a right triangle with rational number side lengths. In other words, a positive integer \( n \) is called a congruent number if there exists a right triangle with legs of rational lengths such that the area of the triangle is equal to \( n \).
A covering system is a concept in mathematics, particularly in the field of number theory and combinatorial number theory. It involves the use of sets of integers or numbers to cover or fill up certain properties or conditions. Specifically, a covering system typically refers to a collection of sets of integers (or natural numbers) such that every integer belongs to at least one of the sets in that collection.
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.
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, ...
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.
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.
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 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.
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.
Firoozbakht's conjecture is a mathematical conjecture related to the properties of prime numbers, specifically concerning the gaps between consecutive prime numbers.
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.
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 \( p_1, p_2, p_3, \ldots \).
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.
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 \(3 + 5\).
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.
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.
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.
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.
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.
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.
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.
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.
Lehmer's totient problem is an unsolved problem in number theory, specifically related to the Euler's totient function \( \phi(n) \). The Euler's totient function \( \phi(n) \) counts the number of positive integers up to \( n \) that are coprime to \( n \).
Lemoine's conjecture, also known as the "Lemoine's problem" or "Lemoine's hypothesis," is a statement in number theory that relates to the representation of numbers as sums of prime numbers. Specifically, it posits that every odd integer greater than 5 can be expressed as the sum of an odd prime and an even prime (which can only be 2).
Leopoldt's conjecture is a conjecture in the field of number theory, particularly concerning \( p \)-adic numbers and the study of class fields. Specifically, it deals with the behavior of abelian extensions of number fields in relation to their \( p \)-adic completions and \( p \)-adic class groups.
The Manin conjecture, proposed by Yuri Manin in the 1970s, is a conjecture in the field of arithmetic geometry. Specifically, it relates to the study of rational points on algebraic varieties, particularly Fano varieties, which are a special class of projective algebraic varieties with ample anticanonical bundles.
Mersenne conjectures typically refer to conjectures related to Mersenne primes, which are prime numbers of the form \( M_n = 2^n - 1 \), where \( n \) is a positive integer. These numbers are named after the French monk Marin Mersenne, who studied them in the early 17th century.
The Minimum Overlap Problem typically refers to a scenario in optimization and scheduling where the goal is to minimize the overlap of certain events, tasks, or processes. This concept can be applied in various fields such as computer science, operations research, and project management, among others. Here are a few specific contexts in which the Minimum Overlap Problem might arise: 1. **Scheduling Tasks**: When scheduling multiple tasks or jobs, it is often desirable to minimize the overlapping of their execution times.
Newman's conjecture is a proposed mathematical conjecture concerning the distribution of the digits in the decimal expansion of the reciprocals of certain integers. More specifically, it relates to the behavior of the leading digits of the decimal expansion of the fractions formed by taking the reciprocal of integers. The conjecture states that for a given positive integer \( n \), the reciprocal \( \frac{1}{n} \) has a certain predictable pattern in the distribution of its leading digits.
Odd Greedy Expansion is a concept used in the realm of algorithms and data structures, particularly in the context of computational problems like Tree Decomposition and dynamic programming on trees. The term is not widely recognized as a standalone concept in mainstream literature but may refer to specific techniques or approaches within graph theory or optimization. In general, a greedy algorithm is one that makes a series of choices, each of which looks best at the moment, with the hope that the overall outcome will be optimal.
Oppermann's conjecture, proposed by mathematician Frank Oppermann in 2012, is a conjecture about the existence of certain types of prime numbers known as "twin primes." Specifically, it suggests that for every positive integer \( n \), there exists a prime number \( p \) such that both \( p \) and \( p + n \) are primes.
A Pierpont prime is a type of prime number that can be expressed in the form \( P = 3 \cdot 2^n + 1 \) or \( P = 2^n + 1 \) for non-negative integers \( n \). The set of Pierpont primes includes numbers generated by these formulas.
Polignac's conjecture, also known as the "French conjecture," is a statement in number theory formulated by the French mathematician Alphonse de Polignac in 1849. The conjecture posits that for every positive even integer \( k \), there are infinitely many prime pairs \( p \) and \( p + k \) such that both \( p \) and \( p + k \) are prime numbers.
A prime triplet refers to a set of three prime numbers that are all two units apart from each other. The most common form of a prime triplet can be expressed as \( (p, p+2, p+6) \) or \( (p-2, p, p+2) \), where \( p \) is a prime number.
Problems involving arithmetic progressions (AP) typically revolve around sequences of numbers in which the difference between consecutive terms is constant. This common difference is a key characteristic of an arithmetic progression.
In number theory, a prime number \( p \) is called a **regular prime** if it does not divide the numerator of the binomial coefficients \( \binom{n}{k} \) for any integers \( n \) and \( k \) where both \( k \) and \( n-k \) are less than \( p \). In simpler terms, a regular prime is one that behaves "nicely" with respect to these combinatorial quantities.
Safe primes and Sophie Germain primes are specific types of prime numbers with particular mathematical properties. ### Sophie Germain Primes A prime number \( p \) is called a *Sophie Germain prime* if \( 2p + 1 \) is also a prime number. In other words, for a prime number \( p \), if \( q = 2p + 1 \) is also prime, then \( p \) is classified as a Sophie Germain prime.
Schanuel's conjecture is a conjecture in transcendental number theory proposed by Stephen Schanuel in the 1960s. It provides a statement about the transcendence of certain numbers related to algebraic numbers and transcendental numbers.
Schinzel's Hypothesis H is a conjecture in number theory proposed by mathematician Andrzej Schinzel in the 1950s. It relates to the distribution of prime numbers generated by certain types of polynomial expressions. Specifically, Schinzel's Hypothesis H deals with a finite collection of multivariable integer polynomials.
A sexy prime is a type of prime number that is part of a pair of primes that have a difference of six. In other words, two prime numbers \( p \) and \( q \) are considered sexy primes if \( q - p = 6 \). For instance, (5, 11) and (7, 13) are examples of sexy prime pairs because both pairs consist of prime numbers that differ by six.
In algebraic number theory, supersingular primes are particularly interesting as they relate to the study of elliptic curves and the behavior of their reduction modulo prime numbers. Specifically, a prime \( p \) is called **supersingular** for an elliptic curve \( E \) over the finite field \( \mathbb{F}_p \) if the reduced curve \( E \mod p \) has a specific structure that makes it "supersingular.
Twin primes are pairs of prime numbers that have a difference of two. In other words, if \( p \) and \( p+2 \) are both prime numbers, then they are considered twin primes. For example, (3, 5), (11, 13), and (17, 19) are all pairs of twin primes. The concept of twin primes is an interesting area of study in number theory.
Vojta's conjecture is a conjecture in the field of arithmetic geometry, named after the mathematician Paul Vojta. It deals with the distribution of rational points on algebraic varieties and is closely related to Diophantine geometry, which studies solutions to polynomial equations. In simple terms, Vojta's conjecture can be thought of as a generalization of the Zsigmondy theorem and the Bombieri-Lang conjecture.
A Wagstaff prime is a special type of prime number that is defined in a particular form. Specifically, a Wagstaff prime is a prime number of the form: \[ \frac{2^p + 1}{3} \] where \( p \) is also a prime number.
The Wall–Sun–Sun prime is a special type of prime number that is defined in relation to a specific type of sequence known as the Fibonacci sequence. A Wall–Sun–Sun prime is a prime number that can be expressed in the form \( F_{k+1} - 1 \), where \( F_k \) is the \( k \)-th Fibonacci number.
The Waring–Goldbach problem is a question in number theory that is an extension of Waring's problem. Specifically, it concerns the representation of even integers as sums of prime numbers. The statement of the problem can be framed as follows: For every even integer \( n \), is there a way to express \( n \) as a sum of a bounded number of prime numbers?
A Wieferich prime is a special type of prime number that satisfies a particular congruence relation involving powers of 2.
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