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Sequence of equally spaced numbers
An arithmetic progression, arithmetic sequence or linear sequence is a sequence of numbers such that the difference from any succeeding term to its preceding
Arithmetic_progression
Theorem on the number of primes in arithmetic sequences
arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Set of prime numbers linked by a linear relationship
primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example
Primes in arithmetic progression
Primes_in_arithmetic_progression
positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form { b , b + a , b + 2 a , . . . } {\displaystyle
Arithmetic progression topologies
Arithmetic_progression_topologies
Type of numeric sequence
mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple
Generalized arithmetic progression
Generalized_arithmetic_progression
On the existence of arithmetic progressions in subsets of the natural numbers
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the
Roth's theorem on arithmetic progressions
Roth's_theorem_on_arithmetic_progressions
Square of numbers with equal row, column and diagonal totals
of s arithmetic progressions with the same common difference among r terms, such that r × s = n2, and whose initial terms are also in arithmetic progression
Magic_square
Property of large sets
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states
Erdős conjecture on arithmetic progressions
Erdős_conjecture_on_arithmetic_progressions
Mathematical sequence of numbers
yields a geometric progression, while taking the logarithm of each term in a geometric progression yields an arithmetic progression. The relation that
Geometric_progression
Progression formed by taking the reciprocals of an arithmetic progression
mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known
Harmonic progression (mathematics)
Harmonic_progression_(mathematics)
Integer side lengths of a right triangle
integers x < y < z {\displaystyle x<y<z} , their squares are in arithmetic progression if z 2 − y 2 = y 2 − x 2 , {\displaystyle z^{2}-y^{2}=y^{2}-x^{2}
Pythagorean_triple
Characterization of how many integers are prime
Erdős–Selberg argument". Let πd,a(x) denote the number of primes in the arithmetic progression a, a + d, a + 2d, a + 3d, ... that are less than x. Dirichlet and
Prime_number_theorem
Number divisible only by 1 and itself
19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes. Many mathematicians
Prime_number
Topics referred to by the same term
Look up progression in Wiktionary, the free dictionary. Progression may refer to: In mathematics: Arithmetic progression, a sequence of numbers such that
Progression
Positional game
The arithmetic progression game is a positional game where two players alternately pick numbers, trying to occupy a complete arithmetic progression of
Arithmetic_progression_game
Long dense subsets of the integers contain arbitrarily large arithmetic progressions
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured
Szemerédi's_theorem
Subset of mathematical connundrums
Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points
Problems involving arithmetic progressions
Problems_involving_arithmetic_progressions
Theorem about prime numbers
arbitrarily long arithmetic progressions. In other words, for every natural number k {\displaystyle k} , there exist arithmetic progressions of primes with
Green–Tao_theorem
Expression for sums of powers
bases in arithmetic progression". Academia.edu. Bazsó, András; Mező, István (2015). "On the coefficients of power sums of arithmetic progressions". Journal
Faulhaber's_formula
Progression-free set of numbers
in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets
Salem–Spencer_set
Spacing between equally-spaced square numbers
numbers in an arithmetic progression of three squares. The congruum problem is the problem of finding squares in arithmetic progression and their associated
Congruum
Type of average of a collection of numbers
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum of a collection
Arithmetic_mean
Integer in Ramsey theory
one of r different colors, then there are at least k integers in arithmetic progression all of the same color. The smallest such N is the van der Waerden
Van_der_Waerden_number
Branch of pure mathematics
branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties
Number_theory
Mathematical sequence involving arithmetic progressions
avoid arithmetic progressions. If S {\displaystyle S} is a finite set of non-negative integers on which no three elements form an arithmetic progression (that
Stanley_sequence
Theorem in Ramsey theory
the same color form an arithmetic progression. But you can't add a ninth integer to the end without creating such a progression. If you add a red 9, then
Van_der_Waerden's_theorem
Mathematical sequence satisfying a specific pattern
multiplication of the elements of a geometric progression with the corresponding elements of an arithmetic progression. The nth element of an arithmetico-geometric
Arithmetico-geometric sequence
Arithmetico-geometric_sequence
On the approximate structure of sets whose sumset is small
then A {\displaystyle A} can be contained in a small generalized arithmetic progression. If A {\displaystyle A} is a finite subset of Z {\displaystyle \mathbb
Freiman's_theorem
Points with no three in a line
subset S ⊂ F p n {\displaystyle S\subset F_{p}^{n}} that contains no arithmetic progression of length 3 {\displaystyle 3} has size at most c p n {\displaystyle
Cap_set
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
Unsolved problem in mathematics
squares, and the middle terms of the three arithmetic progressions themselves form an arithmetic progression? Do there exist three rational right triangles
Magic_square_of_squares
Addition of several numbers or other values
_{i=0}^{n}i=\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\qquad } (Sum of the simplest arithmetic progression, consisting of the first n natural numbers.) ∑ i = 1 n ( 2 i − 1
Summation
Mathematical theorem
on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p(a,d) the least prime in the arithmetic progression a
Linnik's_theorem
Mathematical conjecture
mathematics What is the upper bound for the number of squares in finite arithmetic progressions? More unsolved problems in mathematics Rudin's conjecture is a
Rudin's_conjecture
Repeatable pattern of differences between prime numbers
k-tuple of the form (0, n, 2n, 3n, …, (k − 1)n) is said to be a prime arithmetic progression. In order for such a k-tuple to meet the admissibility test, n must
Prime_k-tuple
Mathematical subject
prime numbers contains arbitrarily long arithmetic progressions. In other words, there exist arithmetic progressions of primes, with k terms, where k can
Arithmetic_combinatorics
Natural number
be the only minimal difference greater than 1 of any increasing arithmetic progression of n primes (in this case, n = 7) that is not a primorial (a product
150_(number)
Type of data visualization for geographic regions
Geometric progression rule divides the range of values so the ratio of thresholds is constant (rather than their interval as in an arithmetic progression). For
Choropleth_map
Australian and American mathematician (born 1975)
mathematicians. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way: In
Terence_Tao
Exploring properties of the integers with complex analysis
L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the
Analytic_number_theory
Right triangle with a feature making calculations on the triangle easier
an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the progression then
Special_right_triangle
Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression. Let π ( x ; q , a ) {\displaystyle \pi (x;q,a)} count the number
Brun–Titchmarsh_theorem
Number of integers coprime to and less than n
distribution of the values of φ ( n ) {\displaystyle \varphi (n)} in the arithmetic progressions modulo q {\displaystyle q} for any integer q > 1 {\displaystyle
Euler's_totient_function
Mathematical process for constructing magic squares
the sum of the arithmetic progression used divided by the order of the magic square. It is possible not to start the arithmetic progression from the middle
Siamese_method
Rational right triangles cannot have square area
If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
"arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers
Arbitrarily_large
Scale of temperature
temperatures), which he labels by two systems, one in arithmetic progression and the other in geometric progression, as follows: Outline of metrology and measurement
Newton_scale
Infinitely many prime numbers exist
Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: that every integer has a unique prime factorization. What Euler wrote
Euclid's_theorem
In number theory, a limitation of sieve theory
in a given arithmetic progression, for example 6 m + 1 {\displaystyle 6m+1} , m = 1 , 2 , … {\displaystyle m=1,2,\dots } or the progression k m + l {\displaystyle
Parity_problem
Ancient algorithm for generating prime numbers
find all of the smaller primes. It may be used to find primes in arithmetic progressions. Sift the Two's and Sift the Three's: The Sieve of Eratosthenes
Sieve_of_Eratosthenes
Triangles without a right angle
equal to 12. The smallest-perimeter triangle with integer sides in arithmetic progression, and the smallest-perimeter integer-sided triangle with distinct
Acute_and_obtuse_triangles
Treatise by Claudius Ptolemy
as experimentally derived, appear to have been obtained from an arithmetic progression. However, according to Mark Smith, Ptolemy's tables were based on
Optics_(Ptolemy)
Number, approximately 3.14
complex numbers at which exp z is equal to one is then an (imaginary) arithmetic progression of the form: { … , − 2 π i , 0 , 2 π i , 4 π i , … } = { 2 π k i
Pi
Triangle with integer side lengths
(see Pythagorean triple). If the angles of any triangle form an arithmetic progression then one of its angles must be 60°. For integer triangles the remaining
Integer_triangle
British mathematician (born 1977)
collaborator Terence Tao, states that there exist arbitrarily long arithmetic progressions in the prime numbers: this is now known as the Green–Tao theorem
Ben_Green_(mathematician)
On the distribution of prime numbers in arithmetic progressions
conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter
Elliott–Halberstam_conjecture
Irrational number based on primes
is irrational; this can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's
Copeland–Erdős_constant
a line? Rudin's conjecture on the number of squares in finite arithmetic progressions The sunflower conjecture – can the number of k {\displaystyle k}
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical proof technique using contradiction
classical interest (for example, the problem of four perfect squares in arithmetic progression). In some cases, to the modern eye, his "method of infinite descent"
Proof_by_infinite_descent
Mathematics award
Infinitely Small Quantities in Leibniz's Mathematics: The Case of his Arithmetical Quadrature of Conic Sections and Related Curves". In Goldenbaum, Ursula;
Fields_Medal
primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions. Define
Siegel–Walfisz_theorem
contains 33 verses covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA)[clarification needed]
List of publications in mathematics
List_of_publications_in_mathematics
Class of ranked-choice electoral systems
valid progression of points or weightings may be chosen at will (Eurovision Song Contest) or it may form a mathematical sequence such as an arithmetic progression
Positional_voting
Prime number of the form 2^n – 1
Mersenne primes, much of which is now done using distributed computing. Arithmetic modulo a Mersenne number is particularly efficient on a binary computer
Mersenne_prime
Prime number p where p+2 is prime or semiprime
many arithmetic progressions of length 3. Binbin Zhou generalized this result by showing that the Chen primes contain arbitrarily long arithmetic progressions
Chen_prime
{\displaystyle a^{2}} , and c 2 {\displaystyle c^{2}} should form an arithmetic progression. That is, b 2 − k = a 2 {\displaystyle b^{2}-k=a^{2}} , and a 2
Automedian_triangle
Infinite sequence of numbers satisfying a linear equation
, … {\displaystyle 0,1,4,9,16,25,\ldots } . All arithmetic progressions, all geometric progressions, and all polynomials are constant-recursive. However
Constant-recursive_sequence
ending in the decimal digit d. If a and d are relatively prime, the arithmetic progression contains infinitely many primes. 2n+1: 3, 5, 7, 11, 13, 17, 19,
List_of_prime_numbers
Mathematical conjecture about zeros of L-functions
on arithmetic progressions: Let π ( x , a , d ) {\textstyle \pi (x,a,d)} where a and d are coprime denote the number of prime numbers in arithmetic progression
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Area of combinatorics in mathematics
partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for |A + B| in
Additive_combinatorics
Numbers whose prime factors all divide the number more than once
are k-powerful numbers in an arithmetic progression. Moreover, if a1, a2, ..., as are k-powerful in an arithmetic progression with common difference d, then
Powerful_number
Number raised to the third power
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number
Cube_(algebra)
Meromorphic function on the complex plane
prime number theorem, according to which in every arithmetic sequence (also called an arithmetic progression) a , a ± n , a ± 2 n , a ± 3 n , … , with gcd
L-function
Theorem in arithmetic combinatorics
result is that any set with small additive doubling (such as an arithmetic progression) has the lower bound on the product set |AA| = Ω(|A|2 log−1(|A|))
Erdős–Szemerédi_theorem
Paradox related to increasing roadway capacity
edge, the energy is the sum of an arithmetic progression, and using the formula for the sum of an arithmetic progression, one can show that E ( Z ) ≤ T (
Braess's_paradox
Branch of mathematical combinatorics
are coloured with c different colours, then it must contain an arithmetic progression of length n whose elements are all the same colour. Hales–Jewett
Ramsey_theory
List of mathematical contexts in which exponentiated terms are summed
coefficients in the polynomial. The sum of cubes of numbers in arithmetic progression is sometimes another cube. The Fermat cubic, in which the sum of
Sums_of_powers
Hungarian mathematician (1913–1996)
on arithmetic progressions: If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary
Paul_Erdős
Triangle whose side lengths and area are integers
internal angles form an arithmetic progression. This is because all plane triangles with interior angles in an arithmetic progression must have one interior
Heronian_triangle
Prime such that p^2 divides 2^(p-1)-1
Quadruplet (p, p + 2, p + 6, p + 8) Cousin (p, p + 4) Sexy (p, p + 6) Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) Balanced (consecutive p − n, p, p + n)
Wieferich_prime
shown in for instance the Akhmim wooden tablets. Knowledge of arithmetic progressions is also evident from the mathematical sources. Clagett, Marshall
Egyptian_algebra
Amount a stimulus must be changed to be detected
illustration of the Weber–Fechner law: Circles in the upper row grow in arithmetic progression: each one is larger by 10 units than previous one. They make an
Just-noticeable_difference
Mathematical concept
− N {\displaystyle X-N} of X {\displaystyle X} . A generalised arithmetic progression in Z {\displaystyle \mathbb {Z} } is a subset in Z {\displaystyle
Approximate_group
The zeros of a linear recurrence relation mostly form a regularly repeating pattern
many full arithmetic progressions, where an infinite arithmetic progression is full if there exist integers a and b such that the progression consists
Skolem–Mahler–Lech_theorem
Set of integers whose sum of reciprocals diverges
equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form an + b with a ≥ 1, b ≥ 1
Large_set_(combinatorics)
Formula whose values are the prime numbers
August 2025 The AP27 is listed in "Jens Kruse Andersen's Primes in Arithmetic Progression Records page" Rowland 2008. Jones et al. 1976. Matiyasevich 1999
Formula_for_primes
Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression –
List_of_real_analysis_topics
S-shaped curve
30 November 1844).", p. 1. Verhulst first refers to arithmetic progression and geometric progression, and refers to the geometric growth curve as a logarithmic
Logistic_function
Triangle with specific characteristics
OEIS. In a Brahmagupta triangle the side lengths form an integer arithmetic progression with a common difference 1. A generalized Brahmagupta triangle is
Brahmagupta_triangle
Open subset of the real–number line
powers of a fundamental length, the complex dimensions appear in an arithmetic progression parallel to the imaginary axis, and are called lattice fractal strings
Fractal_string
Term in the mathematical theory of special functions
symbol and gamma function in that they can be related to a general arithmetic progression in the same manner as those are related to the sequence of consecutive
Pochhammer_k-symbol
Counts pieces of a disk cut by lines
+n).} This can be simplified, using the formula for the sum of an arithmetic progression: f ( n ) = 1 + n ( n + 1 ) 2 = n 2 + n + 2 2 . {\displaystyle f(n)=1+{\frac
Lazy_caterer's_sequence
in arithmetic progressions shows that there are an infinity of primes in each co-prime residue class, and the prime number theorem for arithmetic progressions
Multiplicative_number_theory
About simultaneous modular congruences
in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family. The existence and the uniqueness
Chinese_remainder_theorem
Norwegian mathematician (1917–2007)
which he proved by elementary means the theorems for primes in arithmetic progression and the density of primes. This challenged the widely held view
Atle_Selberg
Treatise by Thomas Malthus
increasing in geometric progression (so as to double every 25 years) while food production increased in an arithmetic progression, which would leave a difference
An Essay on the Principle of Population
An_Essay_on_the_Principle_of_Population
Concept in number theory
the set of all even numbers, then d(A) = 0.5. Similarly, for any arithmetical progression A = { a n + b : n ∈ N } {\displaystyle A=\{an+b:n\in \mathbb {N}
Natural_density
Result on density of prime numbers
Quadruplet (p, p + 2, p + 6, p + 8) Cousin (p, p + 4) Sexy (p, p + 6) Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) Balanced (consecutive p − n, p, p + n)
Bertrand's_postulate
alternating group (Alt(n) is also written as An.) A.M. – arithmetic mean. AP – arithmetic progression. arccos – inverse cosine function. arccosec – inverse
List of mathematical abbreviations
List_of_mathematical_abbreviations
Type of unsustainable business model
ever receive. Since matrix schemes follow the same laws of geometric progression as pyramids, they are subsequently as doomed to collapse. Such schemes
Pyramid_scheme
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
Girl/Female
Hindu
Hard worker to reach goal
Boy/Male
Celtic
Mythical father of Clust.
Girl/Female
Arabic, Indian, Modern, Muslim, Pakistani
Queen
Boy/Male
Hindu, Indian, Marathi
A Cow Herd
Boy/Male
Hindu, Indian
Very Cold as Moon
Girl/Female
Hindu, Indian
Awesome
Boy/Male
Hindu
Intoxicating
Surname or Lastname
English
English : topographic name for someone who lived in a stretch of open country by a wood, or (as a later formation) someone who lived near a field by a wood, from Middle English wode ‘wood’ (Old English wudu) + feld ‘open country’, later with the modern meaning ‘field’.Scottish : habitational name from Woodfield, a place near Annan in Dumfriesshire. A certain Roger Wodyfelde is recorded as holding land in Dumfries in 1365.
Boy/Male
Hindu, Indian, Punjabi, Sikh
Earth; Pledge Keeper
Girl/Female
Hindu, Indian, Tamil
Simple; Beautiful
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
ARITHMETIC PROGRESSION
n.
Arithmetic.
a.
Sexagesimal, or made on the scale of 60; as, logistic, or sexagesimal, arithmetic.
a.
Having equal differences; as, the terms of arithmetical progression are equidifferent.
n.
A system of arithmetic, in which numbers are expressed in a scale of 60; logistic arithmetic.
v. t.
To subtract by arithmetical operation; to deduct.
a.
Of or pertaining to a unit or units; relating to unity; as, the unitary method in arithmetic.
n.
That part of arithmetic which treats of adding numbers.
n.
One skilled in arithmetic.
adv.
Conformably to the principles or methods of arithmetic.
n.
The science of numbers; the art of computation by figures.
a.
Of or pertaining to arithmetic; according to the rules or method of arithmetic.
v. t.
To subject to arithmetical division.
n.
The four "liberal arts," arithmetic, music, geometry, and astronomy; -- so called by the schoolmen. See Trivium.
n.
A book containing the principles of this science.
n.
Arithmetical subtraction.
v. i.
To use figures in a mathematical process; to do sums in arithmetic.
adv.
The arithmetical character 0; a cipher. See Cipher.
v. i.
To perform the arithmetical operation of addition; as, he adds rapidly.
a.
Having an assignable arithmetical or numerical value or meaning; not imaginary.
n.
Regular or proportional advance in increase or decrease of numbers; continued proportion, arithmetical, geometrical, or harmonic.