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Numbers whose differences are not squares
In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy
Square-difference-free_set
state that the number of elements in a square-difference-free set of positive integers could only exceed the square root of its largest value by a polylogarithmic
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Set of the elements not in a given subset
termed the set difference of B and A, written B ∖ A , {\displaystyle B\setminus A,} is the set of elements in B that are not in A. If A is a set, then the
Complement_(set_theory)
British mathematician
best-known bound for square-difference-free sets, showing that a set A ⊂ [ N ] {\displaystyle A\subset [N]} with no square difference has size at most N
Thomas_Bloom
British mathematician (born 1987)
best-known bound for square-difference-free sets, showing that a set A ⊂ [ N ] {\displaystyle A\subset [N]} with no square difference has size at most N
James_Maynard_(mathematician)
Number without repeated prime factors
In mathematics, a square-free integer (or squarefree integer) is an integer that is divisible by no square number other than 1. That is, its prime factorization
Square-free_integer
Mathematical set formed from two given sets
C]\cup [(A\cap B)\times (C\cup D)]\cup [(B\setminus A)\times D]} For the set difference, we also have the following identity: ( A × C ) ∖ ( B × D ) = [ A ×
Cartesian_product
In combinatorics, a square-free word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form XX, where X is
Square-free_word
Mathematical set containing no elements
the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories
Empty_set
Set of elements common to all of some sets
Naive set theory – Informal set theories Symmetric difference – Elements in exactly one of two sets Union – Set of elements in any of some sets "Intersection
Intersection_(set_theory)
Metric for difference between two colors
In color science, color difference or color distance is the separation between two colors. This metric allows quantified examination of a notion that formerly
Color_difference
Paradox in set theory
Zermelo set theory developed into the standard Zermelo–Fraenkel set theory (commonly known as ZFC when including the axiom of choice). The main difference between
Russell's_paradox
Standard system of axiomatic set theory
century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial
Zermelo–Fraenkel_set_theory
Set of elements in any of some sets
sequence List of set identities and relations – Equalities for combinations of sets Naive set theory – Informal set theories Symmetric difference – Elements
Union_(set_theory)
Collection of mathematical objects
{\mathcal {S}}}A=X\cup Y} . The set difference of two sets A {\displaystyle A} and B {\displaystyle B} , is a set, denoted A ∖ B {\displaystyle
Set_(mathematics)
Axiom of set theory
this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory
Axiom_of_choice
Proposition in mathematical logic
specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose
Continuum_hypothesis
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Identities and relationships involving sets
{\displaystyle A\cap (A\cup B)=A} Intersection can be expressed in terms of set difference: A ∩ B = A ∖ ( A ∖ B ) {\displaystyle A\cap B=A\setminus (A\setminus
Algebra_of_sets
Mathematical set that can be enumerated
Definitions vary and care is needed respecting the difference with recursively enumerable. A set S {\displaystyle S} is countable if: Its cardinality
Countable_set
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can
Universal_set
Size of a possibly infinite set
measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set A {\displaystyle A} is generally
Cardinal_number
Collection of sets in mathematics that can be defined based on a property of its members
example, in the proof that there is no free complete lattice on three or more generators. The paradoxes of naive set theory can be explained in terms of
Class_(set_theory)
Branch of mathematics that studies sets
4} is the set {2, 3}. Set difference of U and A, denoted U ∖ A, is the set of all members of U that are not members of A. The set difference {1, 2, 3}
Set_theory
Maximal proper filter
ultrafilters that are not principal are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which
Ultrafilter_on_a_set
Mathematical set of all subsets of a set
considered with the operation of symmetric difference (with the empty set as the identity element and each set being its own inverse), and a commutative
Power_set
Finite collection of distinct objects
finite sets to be countable.) The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union
Finite_set
Infinite set that is not countable
mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related
Uncountable_set
Mathematical concept
use the least real from the r sequence that does not have a rational difference with any element thus far constructed in the v sequence. Continue until
Transfinite_induction
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
3-volume treatise on mathematics, 1910–1913
the notation and terminology has changed. The most obvious difference between PM and set theory is that in PM all objects belong to one of a number of
Principia_Mathematica
Comprehensive list of Magic: The Gathering card sets since its inception in 1993
specific set or block, while compilations are free to pick and choose cards from any set. All expansion sets, and all editions of the base set from Sixth
List of Magic: The Gathering sets
List_of_Magic:_The_Gathering_sets
Informal set theories
complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written
Naive_set_theory
Any one of the distinct objects that make up a set in set theory
mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four
Element_of_a_set
Product of a number by itself
defined as the difference x i − x ¯ {\displaystyle x_{i}-{\overline {x}}} . These deviations are squared, then a mean is taken of the new set of numbers (each
Square_(algebra)
Diagram that shows all possible logical relations between a collection of sets
Intersection of two sets A ∩ B {\displaystyle ~A\cap B} Union of two sets A ∪ B {\displaystyle ~A\cup B} Symmetric difference of two sets A △ B {\displaystyle
Venn_diagram
One-to-one correspondence
bijective; its inverse is the positive square root function. By Schröder–Bernstein theorem, given any two sets X and Y, and two injective functions f:
Bijection
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every
Computable_set
Size of a set in mathematics
the set of square numbers is countable, which was considered paradoxical for hundreds of years before modern set theory (cf. § Pre-Cantorian set theory)
Cardinality
Template that specifies one or more axioms
require that a term be free for a variable in a formula, that a variable occur free in a formula, or that a variable not occur free in a specified formula
Axiom_schema
Methodic assignment of colors to elements of a graph
non-negative integers as the "colors". In general, one can use any finite set as the "color set". The nature of the coloring problem depends on the number of colors
Graph_coloring
Proof by Alan Turing
small canonical set, one of three similar to this: {qi Sj Sk R ql} e.g. If machine is executing instruction #qi and symbol Sj is on the square being scanned
Turing's_proof
Every set is smaller than its power set
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle A} , the set of all subsets of
Cantor's_theorem
Student-led demonstrations in China
Francis L.F. (June 2012). "Generational Differences in the Impact of Historical Events: The Tiananmen Square Incident in Contemporary Hong Kong Public
1989 Tiananmen Square protests and massacre
1989_Tiananmen_Square_protests_and_massacre
Class of mathematical set whose elements are all subsets
In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever
Transitive_set
Number, sum of distinct powers of 4
sequences of numbers with no square difference, Imre Z. Ruzsa found a construction for large square-difference-free sets that, like the binary definition
Moser–de_Bruijn_sequence
after proving that the set of points of a square has the same cardinality as that of the points on just an edge of the square: the cardinality of the
Paradoxes_of_set_theory
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
Possible axiom for set theory in mathematics
)=0} is a free abelian group, where E x t {\displaystyle \mathrm {Ext} } is the Ext functor. The existence of a definable well-order of all sets (the formula
Axiom_of_constructibility
Theorem in set theory
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there
Schröder–Bernstein_theorem
Axiomatic set theories based on the principles of mathematical constructivism
equivalent to the classical statement. The difference is that the constructive proofs are harder to find. In set theory, a restriction to the constructive
Constructive_set_theory
Statistical distance measure
with some notable (and useful) differences, including that it is symmetric and it always has a finite value. The square root of the Jensen–Shannon divergence
Jensen–Shannon_divergence
Proof in set theory
infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some
Cantor's_diagonal_argument
(Mathematical) decomposition into a product
x^{4}+1.} If one introduces the non-real square root of –1, commonly denoted i, then one has a difference of squares x 4 + 1 = ( x 2 + i ) ( x 2 − i ) . {\displaystyle
Factorization
Particular class of sets which can be described entirely in terms of simpler sets
defined within L {\displaystyle L} itself by a formula of set theory with no parameters, only the free-variables x {\displaystyle x} and y {\displaystyle y}
Constructible_universe
Mathematical concept for comparing objects
relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only
Equivalence_relation
System of mathematical set theory
The Kripke–Platek set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can
Kripke–Platek_set_theory
Symbolic description of a mathematical object
{\displaystyle f(x)=x^{2}+1} define the function that associates to each number its square plus one. An expression with no variables would define a constant function
Expression_(mathematics)
System of mathematical set theory
of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of
Morse–Kelley_set_theory
Mathematical construction of a set with an equivalence relation
relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity
Setoid
Method for estimating the unknown parameters in a linear regression model
regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values
Ordinary_least_squares
Theorem for proving more complex theorems
com. Retrieved 2019-11-28. Richeson, Dave (2008-09-23). "What is the difference between a theorem, a lemma, and a corollary?". David Richeson: Division
Lemma_(mathematics)
Area of mathematical logic
structure has quantifier elimination, every set definable in a structure is definable by a quantifier-free formula over the same parameters as the original
Model_theory
Set theory concept
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such
Large_cardinal
Pair of mathematical objects
Morse–Kelley set theory makes free use of proper classes. Morse defined the ordered pair so that its projections could be proper classes as well as sets. (The
Ordered_pair
Symbol representing a mathematical object
by the variables p and q and require that the value of the square of p is twice the square of q, which in algebraic notation can be written p2 = 2 q2
Variable_(mathematics)
Paradox in set theory
In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number
Cantor's_paradox
Type of cardinal number in mathematics
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa }
Regular_cardinal
Basic framework of mathematics
Hamkins proposes a more flexible alternative: a set-theoretic multiverse allowing free passage between set-theoretic universes that satisfy the continuum
Foundations_of_mathematics
System of mathematical set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
Class of numerical techniques
finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences.
Finite_difference_method
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be elements
Non-well-founded_set_theory
American financial services company
reported Square as stating that over 7 million people used the Cash App in December 2017. Vox also stated that Cash App was the "No. 1 free finance app"
Block,_Inc.
Axiom in the mathematical field of set theory
set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory
Martin's_axiom
Product of an integer with itself
less than a square (3 = 22 − 1). More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is,
Square_number
System of mathematical set theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring
General_set_theory
Result in mathematics and set theory
lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej Mostowski (1949, theorem 3) and John Shepherdson (1953)
Mostowski_collapse_lemma
Form of mathematical proof
\exists \,a,b\in \mathbb {N} .\,\,n=4a+5b} However, there will be slight differences in the structure and the assumptions of the proof, starting with the
Mathematical_induction
Axioms for the natural numbers
that all computably enumerable sets are diophantine sets, and thus definable by existentially quantified formulas (with free variables) of PA. Formulas of
Peano_axioms
British mathematician (1925–2015)
approximation, Roth made major contributions to the theory of progression-free sets in arithmetic combinatorics and to the theory of irregularities of distribution
Klaus_Roth
Principle in set theory
mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x. It
Axiom_of_adjunction
Problem in computer science
address different issues and exhibit important conceptual and technical differences. Thus, Davis was simply being modest when he said: It might also be mentioned
Halting_problem
Branch of mathematical logic
difference hierarchy of Σ0 3 sets. For a poset P, let MF(P) denote the topological space consisting of the filters on P whose open sets are the sets of
Reverse_mathematics
Basic notion of sameness in mathematics
mathematics: through logic or through set theory. In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property
Equality_(mathematics)
(or there may be a difference in what can be proved where there is no provable difference between their properties). Further, set theory imports concepts
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Root-finding algorithm
particular the x87 instruction set was very slow at the time compared to modern SSE operations. The fast inverse square generates a good approximation
Fast_inverse_square_root
Natural number
magic square. the number of supersingular primes. the smallest positive number that can be expressed as the difference of two positive squares in more
15_(number)
Instrument for displaying time-varying signals
together results in a display of the differences between them, provided neither channel is overloaded. This difference mode can provide a moderate-performance
Oscilloscope
System of mathematical set theory
1929 axiom system, which is closer to Cantorian set theory. The major differences between Cantorian set theory and the 1929 axiom system are classes and
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Ordered listing of items in collection
(where − denotes set difference). Select any element s' ∈ S' and assign f(n − 1) = s' . Continue this process until all elements of the set have been assigned
Enumeration
American actress, producer, and social activist (born 1937)
eds. (November 27, 2012). When We Were Free to Be: Looking Back at a Children's Classic and the Difference It Made (1st ed.). Chapel Hill, North Carolina:
Marlo_Thomas
Structure of a formal language
A formal grammar is a set of symbols and the production rules for rewriting some of them into every possible string of a formal language over an alphabet
Formal_grammar
2013 video game
multiplayer online role-playing game (MMORPG) developed and published by Square Enix. Directed and produced by Naoki Yoshida and released worldwide for
Final_Fantasy_XIV
Mathematician (1845–1918)
January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established
Georg_Cantor
Axiom in set theory
proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set. The axiom of
Axiom_of_global_choice
Concept in set theory
where the cardinality of the set of all atoms is usually larger than the set of all sets. The significance of this difference is evidenced by the fact that
Urelement
Function computable with bounded loops
≥ b then a−b else 0 Minimum(a1, ... an) Maximum(a1, ... an) Absolute difference: | a−b | =def (a ∸ b) + (b ∸ a) ~sg(a): NOT[signum(a)]: If a=0 then 1
Primitive_recursive_function
Natural number
3172, the smallest 6-digit square 101,101 = smallest palindromic Carmichael number 101,723 = smallest prime number whose square is a pandigital number containing
100,000
Financial services provided by Block, Inc.
firm launched Square Market, which allows sellers to create a free online storefront with online payment processing functionality. Square Stand was introduced
Square_(financial_services)
Token in a mathematical or logical formula
the formal structure: There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed
Symbol_(formal)
SQUARE DIFFERENCE-FREE-SET
SQUARE DIFFERENCE-FREE-SET
Male
Icelandic
Icelandic form of Old Norse Freyr, FREY means "lord, master."
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : nickname or status name from Old English frēo ‘free(-born)’, i.e. not a serf.North German : topographic or habitational name from a place named Frede or Frede(n).North German : nickname from a variant of Middle Low German wrēd ‘crooked’.
Male
Swedish
Swedish name derived from Old Norse stúra, STURE means "obstinate."
Surname or Lastname
English
English : variant of Squire.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Surname or Lastname
English
English : variant of Freer 1.French (Frère) : from frère ‘brother’, used as a byname for the younger of two brothers.
Surname or Lastname
English
English : nickname from Middle English freil, frel(i)e ‘frail’, ‘weak’.Possibly an Americanized spelling of German Friel 2.
Surname or Lastname
English
English : patronymic from Squire.
Surname or Lastname
English (mainly southeastern)
English (mainly southeastern) : topographic name for someone who lived near a conspicuous tree, Middle English tre(w).
Male
Swedish
Danish and Swedish form of Old Norse Freyr, FREJ means "lord, master."
Male
English
Short form of English Frederick, FRED means "peaceful ruler."
Boy/Male
Hindu, Indian
Difference
Boy/Male
English American
Shieldbearer.
Male
English
French form of English Stewart, STUART means "house guard; steward." In use by the English and Scottish.
Boy/Male
Italian
Squire.
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Boy/Male
Australian, British, English
In Liberty
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Female
English
English form of Irish BrÃgh, BREE means "force, strength."
Female
English
Anglicized form of Danish Freya, FREA means "lady, mistress."
SQUARE DIFFERENCE-FREE-SET
SQUARE DIFFERENCE-FREE-SET
Boy/Male
Hindu
God
Boy/Male
Dutch
Lives at the oak.
Boy/Male
Hindu, Indian, Tamil
Lord Siva
Boy/Male
Hindu, Indian, Indonesian
Gentle
Boy/Male
Tamil
Saint
Boy/Male
Tamil
Pyarelal | பà¯à®¯à®¾à®°à¯‡à®²à®¾à®²Â
Lord Krishna
Boy/Male
Arabic, Muslim
Distant
Boy/Male
German English
Spear-fortified town.
Boy/Male
Muslim
Lion, Powerful
Boy/Male
Australian, Finnish, French, Hawaiian, Hebrew, Japanese
Shout for Joy; Song of Joy
SQUARE DIFFERENCE-FREE-SET
SQUARE DIFFERENCE-FREE-SET
SQUARE DIFFERENCE-FREE-SET
SQUARE DIFFERENCE-FREE-SET
SQUARE DIFFERENCE-FREE-SET
n.
One who frees, or sets free.
n.
Having the toe square.
superl.
Certain or honorable; the opposite of base; as, free service; free socage.
n.
The act of differing; the state or measure of being different or unlike; distinction; dissimilarity; unlikeness; variation; as, a difference of quality in paper; a difference in degrees of heat, or of light; what is the difference between the innocent and the guilty?
a.
Forming a right angle; as, a square corner.
adv.
Without charge; as, children admitted free.
n.
To make even, so as leave no remainder of difference; to balance; as, to square accounts.
n.
A square; a measure; a rule.
n.
To multiply by itself; as, to square a number or a quantity.
a.
To make free; to set at liberty; to rid of that which confines, limits, embarrasses, oppresses, etc.; to release; to disengage; to clear; -- followed by from, and sometimes by off; as, to free a captive or a slave; to be freed of these inconveniences.
superl.
Not gained by importunity or purchase; gratuitous; spontaneous; as, free admission; a free gift.
v. t.
To cause to differ; to make different; to mark as different; to distinguish.
n.
Hence, anything which is square, or nearly so
a.
Rendering equal justice; exact; fair; honest, as square dealing.
superl.
Privileged or individual; the opposite of common; as, a free fishery; a free warren.
imp. & p. p.
of Square
n.
An instrument having at least one right angle and two or more straight edges, used to lay out or test square work. It is of several forms, as the T square, the carpenter's square, the try-square., etc.
n.
A square piece or fragment.
imp. & p. p.
of Difference
n.
A square. See 1st Squire.