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Listing all imaginary quadratic fields with a given class number
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of
Class_number_problem
In number theory, measure of non-unique factorization
In mathematics, the ideal class group (or class group) of an algebraic number field K {\displaystyle K} is the quotient group J K / P K {\displaystyle
Ideal_class_group
Concept in algebraic number theory
such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the Stark–Heegner
Heegner_number
Complexity class used to classify decision problems
polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer
NP_(complexity)
number fields with class number 1. It is believed that there are infinitely many such number fields, but this has not been proven. The class number of
List of number fields with class number one
List_of_number_fields_with_class_number_one
Number divisible only by 1 and itself
for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy–Littlewood conjecture F predicts
Prime_number
Class in computational complexity theory
the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of
NC_(complexity)
On algebraic independence of logarithms
Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. To simplify notation, let L {\displaystyle
Baker's_theorem
Issue when estimating a probability
In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case
Reference_class_problem
Field (mathematics) generated by the square root of an integer
theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important. For a nonzero square free integer
Quadratic_field
Unsolved problem in computer science
Unsolved problem in computer science If the solution to a problem can be checked in polynomial time, must the problem be solvable in polynomial time? More
P_versus_NP_problem
Set of problems in computational complexity theory
In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly
Complexity_class
Complexity class
complexity class #P (pronounced "sharp P" or, sometimes "number P" or "hash P") is the set of the counting problems associated with the decision problems in the
♯P
Quadratic imaginary number fields with unique factorisation
special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number. Let Q denote the
Stark–Heegner_theorem
rate of rk(N) (see Szemerédi's theorem) Class number problem: are there infinitely many real quadratic number fields with unique factorization? Fontaine–Mazur
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Conjecture on zeros of the zeta function
consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the
Riemann_hypothesis
23 mathematical problems stated in 1900
solved; the withdrawn 24 would also be in this class. The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation of
Hilbert's_problems
Complexity class
The #P-complete problems (pronounced "sharp P complete", "number P complete", or "hash P complete") form a complexity class in computational complexity
♯P-complete
Open problem on 3x+1 and x/2 functions
Unsolved problem in mathematics For even numbers, divide by 2; For odd numbers, multiply by 3 and add 1. With enough repetition, do all positive integers
Collatz_conjecture
Seven mathematical problems with a US$1 million prize for each solution
selected problems span a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory
Millennium_Prize_Problems
Formula in number theory
In number theory, the class number formula relates many important invariants of an algebraic number field to a special value of its Dedekind zeta function
Class_number_formula
Probability of shared birthdays
In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday
Birthday_problem
On the reciprocity law in algebraic number fields
algebraic number field, where k is a power of a prime. The problem was partially solved for abelian extensions by Artin reciprocity and class field theory
Hilbert's_ninth_problem
Study of numbers that are not solutions of polynomials with rational coefficients
of Baker's theorem contained such bounds, solving Gauss' class number problem for class number one in the process. This work won Baker the Fields Medal
Transcendental_number_theory
Brocard's problem Chinese remainder theorem Chinese remainder theorem class field The class field theory concerns abelian extensions of number fields. class number
Glossary_of_number_theory
Type of computational problem
between complexity classes of P, NP, PH, etc, in circuit complexity, and in interactive proof systems. Let R be a search problem, formalised as a binary
Counting_problem_(complexity)
Area of a right triangle with rational-numbered sides
whether a given rational number is a congruent number is called the congruent number problem. As of 2019[update], this problem has not been brought to
Congruent_number
Mathematical problem involving optimal stopping theory
known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. Its solution is also
Secretary_problem
English mathematician (1939–2018)
made significant contributions to several areas in number theory, such as the Gauss class number problem, diophantine approximation, and to Diophantine equations
Alan_Baker_(mathematician)
1798 textbook by Carl Friedrich Gauss
Modern Number Theory, New York, New York: Springer-Verlag, pp. 358–361, ISBN 978-0-387-97329-6 Goldfeld, Dorian (July 1985), "Gauss' Class Number Problem For
Disquisitiones_Arithmeticae
Finite extension of the rationals
{\displaystyle K} has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss
Algebraic_number_field
Problem of determining if a Boolean formula could be made true
is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP, which includes
Boolean satisfiability problem
Boolean_satisfiability_problem
Method for solving one problem using another
second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second and the number of times
Polynomial-time_reduction
Process of achieving a goal by overcoming obstacles
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from
Problem_solving
NP-hard problem in combinatorial optimization
exponentially) with the number of cities. The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It
Travelling_salesman_problem
American mathematician (1950–2025)
the L-function of elliptic curves and led to breakthroughs on the class number problem of Carl Friedrich Gauss. Gross first met his wife Jill P. Mesirov
Benedict_Gross
Odd number with specific properties
set. Unsolved problem in mathematics Is 78,557 the smallest Sierpiński number? More unsolved problems in mathematics The Sierpiński problem asks for the
Sierpiński_number
Physics problem related to laws of motion and gravity
In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses
Three-body_problem
Prize awarded by the American Mathematical Society
doi:10.2307/1970476. Zbl 0223.02050. Goldfeld, Dorian (1985). "Gauss' class number problem for imaginary quadratic fields" (PDF). Bulletin of the American Mathematical
Cole_Prize
Complexity class
algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven
NP-hardness
Generalization of "n-th" to infinite cases
(α + 1)-th number class is the cardinality immediately following that of the α-th number class. For a limit ordinal α, the α-th number class is the union
Ordinal_number
Université de Montréal Gauss map in number theory Gaussian moat Gauss class number problem Gauss's multiplication formula Gaussian period Gaussian rational
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Concept in theoretical computer science
analysis. Definition of the class RTM - Reversal Turing Machines, simple and strong subclass of the TMs. "The Busy Beaver Problem: A New Millennium Attack"
Busy_beaver
Iwaniec, Henryk (2002). "Spacing of zeros of Hecke L-functions and the class number problem". Acta Arithmetica. 103 (3): 259–312. arXiv:math/0111012. Bibcode:2002AcAri
Grand_Riemann_hypothesis
Potential counterexample to the generalized Riemann hypothesis
D | ) − 1 {\textstyle h(D)\gg {\sqrt {|D|}}(\log |D|)^{-1}} (see Class number problem for comparison). This can be extended to an equivalence, as it is
Siegel_zero
Partition of a graph's nodes into cliques
graphs, which are also classes of perfect graphs. The clique cover problem remains NP-complete on some other special classes of graphs, including the
Clique_cover
Type of prime number
{\displaystyle e^{-1/2}} ? More unsolved problems in mathematics In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850
Regular_prime
number of accepting computation paths is odd. An example of a ⊕P problem is "does a given graph have an odd number of perfect matchings?" The class was
Parity_P
Gaussian period Fermat's Last Theorem Class number problem for imaginary quadratic fields Stark–Heegner theorem Heegner number Langlands program Different ideal
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Numbers that evenly divide powers of 60
th power of a polynomial. As with other classes of smooth numbers, regular numbers are important as problem sizes in computer programs for performing
Regular_number
the Sharp Satisfiability Problem (sometimes called Sharp-SAT, #SAT or model counting) is the problem of counting the number of interpretations that satisfy
♯SAT
Combinatorial optimization problem
problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of
Assignment_problem
Subset of a graph's vertices, including at least one endpoint of every edge
cover problem can be formulated as the following integer linear program (ILP). This ILP belongs to the more general class of ILPs for covering problems. The
Vertex_cover
On short connecting nets with added points
the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization
Steiner_tree_problem
Mathematical and computational problem
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each
Bin_packing_problem
Abstract machine used to study decision problems
of a certain problem R {\displaystyle R} in a single operation. The problem R {\displaystyle R} can be of any complexity class, or it can even
Oracle_machine
Subset of a graph's nodes such that all other nodes link to at least one
neighbor in D. The domination number γ(G) is the number of vertices in a smallest dominating set for G. The dominating set problem concerns testing whether
Dominating_set
American mathematician (born 1947)
solution of Gauss's class number problem for imaginary quadratic fields. Specifically, he proved an effective lower bound for the class number of an imaginary
Dorian_M._Goldfeld
Unsolved problem in computational complexity theory
time. There are a number of classes of mathematical objects for which the problem of isomorphism is a GI-complete problem. A number of them are graphs
Graph_isomorphism_problem
Set of objects whose state must satisfy limits
Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations
Constraint satisfaction problem
Constraint_satisfaction_problem
complement problem. For example, one important problem is whether a number is a prime number. Its complement is to determine whether a number is a composite
Complement_(complexity)
Yes/no problem in computer science
natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?" A decision procedure for a decision problem is
Decision_problem
Class of problems in computer science
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability
PP_(complexity)
Inherent difficulty of computational problems
of a problem being hard for a complexity class. A problem X {\displaystyle X} is hard for a class of problems C {\displaystyle C} if every problem in C
Computational complexity theory
Computational_complexity_theory
2009. Retrieved 23 June 2009. Goldfeld, Dorian (July 1985). "Gauss' Class Number Problem For Imaginary Quadratic Fields" (PDF). Bulletin of the American Mathematical
List of publications in mathematics
List_of_publications_in_mathematics
Unsolved problem in extremal graph theory
Unsolved problem in mathematics What is the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete
Zarankiewicz_problem
Methodic assignment of colors to elements of a graph
algorithmic problem since the early 1970s: the chromatic number problem (see section § Vertex coloring below) is one of Karp's 21 NP-complete problems from 1972
Graph_coloring
Problem of finding a cycle through all vertices of a graph
Hamiltonian cycle. The Hamiltonian path problem and the Hamiltonian cycle problem belong to the class of NP-complete problems, as shown in Michael Garey and David
Hamiltonian_path_problem
Problem of inverting exponentiation in groups
known algorithm for solving the discrete logarithm problem in general, the first three steps of the number field sieve algorithm only depend on the group
Discrete_logarithm
1998 short novel by Philibert Schogt
believes he has solved one of the great problems of mathematics: "Beauregard's Wild Number Problem" (a fictitious problem invented by Schogt for the novel)
The_Wild_Numbers
Category of mathematical proof
Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic)
Proof_of_impossibility
Problems which attempt to find the most efficient way to pack objects into containers
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to
Packing_problems
Problem in computer science
In computability theory, the halting problem is the decision problem of, given an arbitrary computer program and an input, determining whether said program
Halting_problem
American mathematician
along with being an ingredient to Dorian Goldfeld's solution of the class number problem. As a part of their work, Gross and Zagier found a formula for norms
Don_Zagier
reducing unlabeled problems to labeled ones: each unlabeled class is considered as a symmetry class of labeled objects. The number of unlabelled graphs
Graph_enumeration
Area of discrete mathematics
arbitrary genus. Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Dénes Kőnig. The
Graph_theory
Figurate number
{T_{n}}{L_{n}}}={\frac {1}{3}}.} The triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with
Triangular_number
Problem in machine learning and statistical classification
classification is the problem of classifying instances into one of three or more classes (classifying instances into one of two classes is called binary classification)
Multiclass_classification
Class of British steam locomotives
Single 2-2-2 class was a type of express passenger locomotive designed by John Ramsbottom. The class is better known as the Problem class for the first
LNWR_Lady_of_the_Lake_Class
Concept in computer science
science, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time
BPP_(complexity)
Class of computational problems
theory, a transcomputational problem is a problem that requires processing of more than 1093 bits of information. Any number greater than 1093 is called
Transcomputational_problem
Every graph has evenly many odd vertices
lemma and the mountain climbing problem the geometric properties of the formula commonly arise. The complexity class PPA encapsulates the difficulty of
Handshaking_lemma
Mathematical problem
The Sleeping Beauty problem, also known as the Sleeping Beauty paradox, is a puzzle in decision theory in which an ideally rational epistemic agent is
Sleeping_Beauty_problem
Problem in physics and celestial mechanics
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally
N-body_problem
Root of a quadratic polynomial with a unit leading coefficient
(see Stark–Heegner theorem). This is a special case of the famous class number problem. There are many known positive integers D > 0 for which the ring
Quadratic_integer
Estimate of time taken for running an algorithm
complexity class P. Cobham's thesis posits that these algorithms are impractical, and in many cases they are. Since the P versus NP problem is unresolved
Time_complexity
Unrelated vertices in graphs
clique problems may be very different when restricted to special classes of graphs. For instance, for sparse graphs (graphs in which the number of edges
Independent set (graph theory)
Independent_set_(graph_theory)
Thought experiment
The Two Generals' Problem appears often as an introduction to the more general Byzantine Generals problem in introductory classes about computer networking
Two_Generals'_Problem
Subset of a graph's edges
cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and
Edge_cover
Problem in finite group theory
decidability of the word problem for the finitely generated group G {\displaystyle G} . The related but different uniform word problem for a class K {\displaystyle
Word_problem_for_groups
German mathematician (1908–1975)
named in his honour. Biography portal Class number problem Deuring–Heilbronn phenomenon Heilbronn triangle problem Heilbronn set Heilbronn Institute for
Hans_Heilbronn
Field in algebra
non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite. Let A = K⟨x1, ..., xn⟩
Golod–Shafarevich_theorem
Describes the objects of a given type, up to some equivalence
object is equivalent to exactly one class. A few issues related to classification are the following. The equivalence problem is "given two objects, determine
Classification_theorem
Deterministic quantum algorithm
exponential number of queries to the black box to solve the problem. More formally, it yields an oracle relative to which EQP, the class of problems that can
Deutsch–Jozsa_algorithm
Concept in commutative algebra
are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic
Excellent_ring
Complexity class of problems
complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such
NP-intermediate
Mathematical study of illumination of rooms with mirrored walls
Illumination problems are a class of mathematical problems that study the illumination of rooms with mirrored walls by point light sources. The original
Illumination_problem
Yes-or-no question that cannot ever be solved by a computer
that, via a specific Gödel numbering, correspond to inputs that satisfy the decision problem's criteria. A decision problem A is called decidable or effectively
Undecidable_problem
Class of problems solvable in polynomial time
as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using
P_(complexity)
Odd number with specific properties
then k is a Sierpiński number. Unsolved problem in mathematics Is 509,203 the smallest Riesel number? More unsolved problems in mathematics In 1956,
Riesel_number
CLASS NUMBER-PROBLEM
CLASS NUMBER-PROBLEM
Girl/Female
English American
Born during the summer.
Surname or Lastname
English
English : from the medieval personal name Classe, a short form of Nicholas. See also Clayson.Variant of Klaas or Klass, North German forms of Claus.
Male
English
English form of Norman Germanic Huncberct, possibly HUMBERT means "bright support."Â
Surname or Lastname
English
English : perhaps a variant of Pamber, a habitational name from a place in Hampshire named Pamber, from Old English penn ‘fold’, ‘enclosure’ + beorg ‘hill’.
Female
English
English name derived from the vocabulary word, summer, from Old English sumor, SUMMER means "summer," the hot season of the year.
Surname or Lastname
English
English : occupational name for a summoner, an official who was responsible for ensuring the appearance of witnesses in court, Middle English sumner, sumnor.William Sumner came to Dorchester, MA, from England in about 1635. His descendants include U.S. Senator Charles Sumner, a major force in the struggle to end slavery, who was born in 1811 in Boston.
Surname or Lastname
English
English : from the medieval female personal name Cass, a short form of Cassandra. This was the name (of uncertain, possibly non-Greek, origin) of an ill-fated Trojan prophetess of classical legend, condemned to foretell the future but never be believed; her story was well known and widely popular in medieval England.
Male
German
Short form of German Niclaus, CLAUS means "victor of the people."Â
Female
Native American
Native American Algonquin name NUMEES means "sister."
Girl/Female
Muslim American Arabic English Gaelic
Jewel. Amber stone.
Surname or Lastname
English and German
English and German : metonymic occupational name for a glazier or glass blower, from Old English glæs ‘glass’ (akin to Glad, referring originally to the bright shine of the material), Middle High German glas.Irish and Scottish : Anglicized form of the epithet glas ‘gray’, ‘green’, ‘blue’ or any of various Gaelic surnames derived from it.German : altered form of the personal name Klass, a reduced form of Nikolaus (see Nicholas).Jewish (Ashkenazic) : ornamental name from German Glass ‘glass’, or a metonymic occupational name for a glazier or glass blower.
Male
German
German byname BAMBER means "short and fat."Â
Female
English
English short form of Latin Cassandra, CASS means "she who entangles men."Â
Surname or Lastname
English
English : nickname from Old French, Middle English cras ‘big’, ‘fat’ (Latin crassus).Possibly an altered spelling of German Krass.
Girl/Female
American, Arabic, Australian, British, Chinese, English, Hebrew
The Warmest Season of the Year; Summer Season; Name of the Season; Summer; The Hot Season of the Year
Boy/Male
Hindu, Indian
Number
Boy/Male
Greek Latin
People's victory.
Surname or Lastname
English
English : habitational name from any of the various places so called from their situation on a stream with this name. Humber is a common prehistoric river name, of uncertain origin and meaning.
Surname or Lastname
English
English : variant of Close 1.German : variant of Kloss.
Surname or Lastname
North German
North German : topographic name from Middle Low German plas ‘place’, ‘open square’, ‘street’.South German (also Pläss) : from a short form of the medieval personal name Blasius.English : variant of Place 3.
CLASS NUMBER-PROBLEM
CLASS NUMBER-PROBLEM
Girl/Female
French Latin
Light; illumination.
Boy/Male
Hebrew
God has made great.
Male
English
Pet form of English Chester, CHET means "soldier's camp."
Boy/Male
Hindu, Indian, Tamil
It's Name of God
Boy/Male
Muslim
Thought, Imagination, Ecstasy, Mirth, Devotion
Boy/Male
Arabic, Muslim
Defiant
Girl/Female
Arabic, Muslim, Pashtun
Honourable
Boy/Male
English Latin
Strong.; the name of more than 50 saints and three Roman emperors.
Female
English
Anglicized form of Irish RóisÃn, ROSHEEN means "little rose."
Boy/Male
Indian
Middle
CLASS NUMBER-PROBLEM
CLASS NUMBER-PROBLEM
CLASS NUMBER-PROBLEM
CLASS NUMBER-PROBLEM
CLASS NUMBER-PROBLEM
v. t.
To case in glass.
v. t.
To shut or fasten together with, or as with, a clasp; to shut or fasten (a clasp, or that which fastens with a clasp).
b. t.
To fill or encumber with lumber; as, to lumber up a room.
v. t.
A looking-glass; a mirror.
n.
To amount; to equal in number; to contain; to consist of; as, the army numbers fifty thousand.
n.
A group of individuals ranked together as possessing common characteristics; as, the different classes of society; the educated class; the lower classes.
n.
A numeral; a word or character denoting a number; as, to put a number on a door.
n.
To divide into classes, as students; to form into, or place in, a class or classes.
v. t.
Variant of Clasp
n.
A number of students in a school or college, of the same standing, or pursuing the same studies.
a.
Of or pertaining to umber; like umber; as, umbery gold.
n.
A brown or reddish pigment used in both oil and water colors, obtained from certain natural clays variously colored by the oxides of iron and manganese. It is commonly heated or burned before being used, and is then called burnt umber; when not heated, it is called raw umber. See Burnt umber, below.
n.
One of the sections into which a church or congregation is divided, and which is under the supervision of a class leader.
n.
Number; -- often abbrev. No.
n.
To arrange in classes; to classify or refer to some class; as, to class words or passages.
a.
Of the rank or degree below the best highest; inferior; second-rate; as, a second-class house; a second-class passage.
n.
To give or apply a number or numbers to; to assign the place of in a series by order of number; to designate the place of by a number or numeral; as, to number the houses in a street, or the apartments in a building.
v. t.
Anything made of glass.
imp. & p. p.
of Number