n
(mathematics) A composite number n that satisfies the modular arithmetic congruence relation bⁿ⁻¹≡1(mod n) for all integers 1
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(mathematics) An integer of the form 4ⁿ-2ⁿ⁺¹-1.
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(mathematics) A Carol number that is also a prime number.
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In combinatorial mathematics, any of a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects; the nᵗʰ Catalan number is equal to (2n choose n) over (n+1).
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(mathematics, countable) Any of various similar theorems.
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(number theory) Any prime number p such that p+2 is either a semiprime or another prime.
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(number theory) A theorem stating that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime.
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(number theory) A theorem stating that, if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
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(mathematics) Any integer that is a common multiple of the denominators of two or more fractions.
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(mathematics) The constant added to each element of an arithmetic progression to obtain the next.
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(mathematics) A number that can be divided into two different numbers, without leaving a remainder.
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(mathematics) A number which may be divided by any of a given set of numbers without a remainder.
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(number theory) A (nonzero) natural number that is expressible as the product of two (or more) natural numbers other than itself and 1.
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(algebra, ring theory, of a polynomial with coefficients in a GCD domain) The greatest common divisor of the coefficients; (of a polynomial with coefficients in an integral domain) the common factor of the coefficients which, when removed, leaves the adjusted coefficients with no common factor that is noninvertible.
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(mathematics) The constant 1.303577269034..., an algebraic number of degree 71, related to the limit of the growth of the look-and-say sequence.
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(mathematics) An irrational constant that is the concatenation of "0." with the base-10 representations of the prime numbers in order, i.e. approximately 0.235711131719232931374143...
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(mathematics, cryptography) A method of finding small integer zeros of univariate or bivariate polynomials modulo a given integer. It forms the basis of Coppersmith's attack.
adj
(algebra, by extension, of two or more polynomials) Whose greatest common divisor is a nonzero constant (i.e., polynomial of degree 0).
n
(mathematics) The value of n – φ(n), i.e. the number of positive integers less than or equal to n that are divisible by at least one prime that also divides n.
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(mathematics) Either of a pair of prime numbers that differ by four
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(number theory) An estimate for the size of gaps between consecutive prime numbers, stating that p_n+1-p_n=O(( log p_n)²), where pₙ denotes the nth prime number, O is big O notation, and log is the natural logarithm.
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(mathematics) A natural number of the form n·2ⁿ+1.
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(mathematics) A Cullen number that is prime.
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A Cunningham chain of the first kind of length n is a sequence of prime numbers (p₁, ..., pₙ) such that for all 1 ≤ i < n, pᵢ₊₁ = 2pᵢ + 1.
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(mathematics) An integer for which cyclic permutations of the digits are successive integer multiples of the number.
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(mathematics) A cyclic sequence (of order n on a size-k alphabet A) in which every possible length-n string on A occurs exactly once as a substring (i.e. a contiguous subsequence). Such a sequence is denoted by B(k, n).
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(mathematics) A theorem about packing congruent rectangular bricks into larger rectangular boxes so that no space is left over. It states that a "harmonic brick" (one in which each side length is a multiple of the next smaller side length) can only be packed into a box whose dimensions are multiples of the brick's dimensions.
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(graph theory) A theorem stating that, when all finite subgraphs of an infinite graph can be colored using c colors, the same is true for the parent graph.
adj
(mathematics) Of a number n, Having the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n).
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(number theory) A number that is greater than the sum of all of its divisors except itself.
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(mathematics) Any member of a certain polynomial sequence Dₙ(x,α).
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(number theory) The conjecture that, for a finite set of linear forms a₁ + b₁n, a₂ + b₂n, ..., aₖ + bₖn with bᵢ ≥ 1, there are infinitely many positive integers n for which they are all prime, unless there is a congruence condition preventing this.
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(number theory) A polynomial equation whose variables are only permitted to assume integer values.
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(set theory) The set of all possible tuples whose elements are elements of given, separately specified, sets.
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(algebra) An expression that gives information about the roots of a polynomial; for example, the expression D = b² - 4ac determines whether the roots of the quadratic equation ax² + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0).
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(number theory, algebra) Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ.
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(mathematics) A function yielding the sum of the divisors of an integer. It is denoted by the Greek letter σ (sigma).
adj
(geometry, of a cycle or divisor) Having no negative coefficients.
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(mathematics) Any Eisenstein integer that is prime.
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(number theory) A prime number that becomes a different prime when its decimal digits (or digits in some specified other base) are reversed.
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(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.
adj
(mathematics, of a family of functions) Having the property that there exists a positive number M such that for all functions f in the family of functions F defined on X, |f(x)| ≤ M for all x in X.
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Alternative form of Erdős number [(mathematics) A number describing the degree of collaboration between a mathematician and Paul Erdős, defined such that a mathematician who has written a paper with Erdős has an Erdős number of 1, a mathematician who has written a paper with somebody having an Erdős number of 1 has an Erdős number of two, and so on.]
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(mathematics) A number describing the degree of collaboration between a mathematician and Paul Erdős, defined such that a mathematician who has written a paper with Erdős has an Erdős number of 1, a mathematician who has written a paper with somebody having an Erdős number of 1 has an Erdős number of two, and so on.
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(algebra, by generalisation) the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.
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(arithmetic, number theory) Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method.
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(number theory) A theorem which states that, given a positive integer a which is coprime to an odd prime number p, a is a quadratic residue of p if and only if a^((p-1)/2) is congruent to 1 modulo p.
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(number theory) The function that counts how many integers below a given integer are coprime to it.
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(mathematics, combinatorics) The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.
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(mathematics) A prime number that is one more or one less than a factorial. For example, 7 is a factorial prime, since 3! + 1 = 7.
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(mathematics) A number that is equal to the sum of the factorials of its digits.
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(number theory) For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.
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(mathematics) A theorem stating that every finite group of odd order is solvable.
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(number theory) An integer which is one more than two raised to a power which is itself a power of two (i.e., is expressible in the form 2^(2ⁿ)+1 for some n>0); equivalently, a number that is one more than two raised to some power (is expressible as 2ⁿ+1) and is prime.
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(number theory) A Fermat number that is prime.
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(mathematics) With respect to an integer base b, with b > 1, a composite integer n such that bⁿ⁻¹ is congruent to one modulo n.
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(number theory) The theorem that the Diophantine equation aⁿ+bⁿ=cⁿ has no solutions for positive integers a,b,c,n, where n>2.
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(number theory) The theorem that, for any prime number p and integer a, aᵖ-a is an integer multiple of p.
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The position (at the 762nd digit after the decimal point) in the decimal expansion of pi at which a sequence of six consecutive nines first appears, unexpectedly early in view of the expected and otherwise apparent randomness of said expansion.
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(mathematics) Any number in a Fibonacci sequence (being the sum of the preceding two numbers)
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(mathematics) Any sequence of numbers such that each is the sum of the preceding two.
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(mathematics, mathematical analysis) Any series resulting from the decomposition of a periodic function into terms involving cosines and sines (or, equivalently, complex exponentials).
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a Fourier series consisting solely of sine functions
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(mathematics) In additive combinatorics, a central result that indicates the approximate structure of sets whose sumset is small. It roughly states that if |A+A|/|A| is small, then A can be contained in a small generalized arithmetic progression.
adj
(mathematics, of a number) Smooth: that factors completely into small prime numbers.
n
For a given set of coprime positive integers, the greatest integer that cannot be expressed as a linear combination (with nonnegative integer coefficients) of its elements.
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(number theory) the theorem that states that every integer greater than one is uniquely expressible as a product of prime numbers, which is called its prime factorization
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(group theory) A theorem which proves that, given a mind swap machine that can't be used on the same pair of people more than once, it is possible to reverse any number of mind swaps by introducing two more people that did not have their minds swapped with anyone before.
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(set theory) A demonstration of a surprising property of infinite sets. Some positive integers are squares while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares; yet for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.
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(mathematics, physics) A theorem about certain vector spaces, relating to string theory.
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(mathematics) A positive even integer that can be expressed as the sum of two odd primes; involved in Goldbach's conjecture.
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(mathematics) The expression of a given even number as the sum of two primes; involved in Goldbach's conjecture.
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(number theory) A conjecture stating that every even integer greater than 2 can be expressed as the sum of two primes, which has been shown to hold up through 4 × 10¹⁸, but remains unproven.
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(mathematics) The problem of generalizing Paul Gordan's computational approach to the theorem of the finiteness of generators for binary forms to work with functions having more than two variables.
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(mathematics) An integer sequence that counts the odd numbers in each row of Pascal's triangle.
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An extremely large number that is an upper bound on the solution to a certain problem in Ramsey theory.
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(arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring).
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Alternative form of greatest common divisor [(arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring).]
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(number theory) A theorem stating that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
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(mathematics) A proposed form of integer notation that would be able to handle arithmetic of infinities
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(mathematics, category theory) A kind of universal set whose elements follow the rules of Zermelo–Fraenkel set theory, and for which, with respect to an arbitrary set, an instance of its kind which has that set as a member may be posited to exist through an additional Tarski–Grothendieck axiom (which is not part of ZF but augments it, yielding Tarski–Grothendieck set theory).
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(geometry) A theorem asserting that every valuation on convex bodies in Rⁿ that is continuous and invariant under rigid motions of Rⁿ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).
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(mathematics) A fundamental combinatorial result of Ramsey theory, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure, and cannot be completely random.
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(mathematics) The number 1729, one of the taxicab numbers.
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(number theory) Any of a series of numbers formed from the sum of the reciprocals of consecutive natural numbers
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(mathematics) A sequence of numbers whose reciprocals form an arithmetic progression.
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(mathematics, mathematical analysis) The divergent series whose terms are the reciprocals of the positive integers; the series ∑ₙ₌₁ ᪲1/n.
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(mathematics) A positive integer which is divisible by the sum of all of its digits.
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(mathematics) A squarefree positive integer d such that the imaginary quadratic field Q(√(−d)) has class number 1; equivalently, such that its ring of integers has unique factorization.
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(geometry) The mathematical problem of determining the set of numbers that can be Heesch numbers.
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(mathematical analysis) A theorem which states that for any subset S of an n-dimensional Euclidean space, S is compact if and only if it is both closed and bounded.
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(mathematics) A positive integer with a half-integral abundancy index
adj
(mathematics) multiperfect such that σ(n) = 11n
n
(mathematics) A result on the class group of certain number fields, strengthening Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity iff p divides the numerator of the n-th Bernoulli number Bₙ for some n, 0 < n < p − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when p divides such an Bₙ.
adj
(mathematics) multiperfect such that σ(n) = 6n
n
Alternative form of greatest common divisor [(arithmetic, number theory) The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring).]
n
Used other than figuratively or idiomatically: see highly, composite number; A positive integer that has a relatively large number of divisors.
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A number that shares its prime factors with another.
adj
(mathematics, number theory) Having the property that, for some positive integer k, δk(n) > 0, where δk(n) = n(k+1) +(k-1) –kσ(n) and σ(n) is the sum of the positive divisors of n.
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(mathematics, number theory) Any natural number n for which, for some positive integer k, n = 1 + k(σ(n) - n - 1), where σ(n) is the sum of the positive divisors of n.
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(set theory) A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.
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(number theory) An algebraic integer that represents an ideal in the ring of integers of a number field.
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(mathematics) A prime number that encodes information that is illegal to possess or distribute.
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(mathematics) The decomposition of a composite number into a product of smaller integers
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(mathematics) The set of all integers; the set {... -3, -2, -1, 0, 1, 2, 3 ...}.
adj
(mathematics) Within a set.
n
(mathematics) A set of natural numbers that contains all finite sums of some infinite set.
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(mathematics) Any odd prime number that is not a regular prime.
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(mathematics) A non-negative integer, the representation of whose square in its base can be split into two parts that add up to itself (such as 297, whose square, 88209, can be split into 88 and 209, totalling 297).
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(mathematics) A repetitive Fibonacci-like integer which appears in a linear recurrence relation.
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(mathematics) A modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9.
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(mathematics) A set of pairs of a mapping's domain which are mapped to the same value.
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A relationship that considers two elements equivalent if one can be obtained from the other by a sequence of transformations of the form yzx -> yxz whenever x zxy whenever x ≤ y < z.
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(mathematics) A unary function, written as δ with a single index, which evaluates to 1 at zero, and 0 elsewhere.
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(number theory) (a/n) or (a|n), a generalization of the Jacobi symbol to all integers n.
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(mathematics) A formula for the exponent of the highest power of a prime number that divides a given binomial coefficient.
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(mathematics) The rule that the limit of the ratio of two functions equals the limit of the ratio of their derivatives, usable when the former limit is indeterminate and the latter limit exists.
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(number theory) The constant of proportionality (approximately 0.7642) in the relationship between the number of positive integers less than x that are the sum of two square numbers, for large x, and the expression x/√.
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(mathematics) Initialism of least common multiple. [(number theory) The smallest positive integer which is divisible by (equivalently, is an integer multiple of) each of a specified finite set of integers.]
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(number theory) The smallest positive integer which is divisible by (equivalently, is an integer multiple of) each of a specified finite set of integers.
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(number theory) A mathematical function of an integer and a prime number, written (a/p), which indicates whether a is a quadratic residue modulo p.
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A number of the form xʸ+yˣ, where x and y are integers greater than 1.
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(mathematics) A particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis.
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(set theory) A cardinal number that cannot be reached from another by repeated successor operations (a weak limit cardinal) or by repeated power set operations (a strong limit cardinal).
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(number theory) A result that is useful in establishing the transcendence of numbers, stating that, if α₁, ..., αₙ are algebraic numbers which are linearly independent over the rational numbers ℚ, then e^(α₁), ..., e^(αₙ) are algebraically independent over ℚ.
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(number theory) An irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that 0<|x-p/q|<1/(qⁿ).
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Any member of an integer sequence that has the same recursive relationship as the Fibonacci sequence, i.e. each term is the sum of the two previous terms, but with different starting values such that the ratios of successive terms approach the golden ratio.
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(mathematical logic) A theorem stating that, if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.
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(mathematics) The number two, as the first Mersenne prime.
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(mathematics) The number three as the second Mersenne prime.
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(physics) The number of neutrons or protons in nuclei which are required to fill the major quantum shells, and thus produce exceptionally stable nuclei: 2, 8, 20, 28, 50, 82 and 126.
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A positive integer x, y or z that is part of a solution to the Markov Diophantine equation x²+y²+z²=3xyz.
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(mathematics) A theorem stating that every computably enumerable set is a Diophantine set, and the converse.
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(mathematics) A prime number with at least one million decimal digits.
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(number theory) A prime number which is one less than a power of two (i.e., is expressible in the form 2ⁿ-1; for example, 31=2⁵-1).
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(mathematics) A prime number whose digital representation contains no smaller subsequence that is also prime.
n
(cryptography) Applied number theory.
adj
(mathematics) Being a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (or k-fold perfect) iff the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn.
n
The sum of a whole number and a number of the form 1/n for n a whole number larger than 1.
adj
(mathematics, of a function, etc.) Distributive over multiplication.
n
(combinatorics) A function from the natural numbers to the set {−1, 0, 1} which maps perfect squares to 0, prime numbers to −1, and is multiplicative.
n
(mathematics) A tuple containing n terms.
n
(set theory) von Neumann-Bernays-Gödel (axiomatic) set theory
n
(algebra) A nilpotent element.
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(mathematics) A natural number whose square cannot be written as the sum of two nonzero squares.
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A number that is not a prime number.
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(mathematics) A positive integer that is divisible by every one of its digits.
adj
(mathematics) multiperfect such that σ(n) = 8n
n
The On-Line Encyclopedia of Integer Sequences
n
(algebra) A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.
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(mathematics) A prime number larger than some given natural number.
n
(number theory, field theory) A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form pᵏ(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p⁻ᵏ and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)
n
(number theory) A p-adic absolute value, for a given prime number p, the function, denoted |..|ₚ and defined on the rational numbers, such that |0|ₚ = 0 and, for x≠0, |x|ₚ = p^(-ordₚ(x)), where ordₚ(x) is the p-adic ordinal of x; the same function, extended to the p-adic numbers ℚₚ (the completion of the rational numbers with respect to the p-adic ultrametric defined by said absolute value); the same function, further extended to some extension of ℚₚ (for example, its algebraic closure).
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(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.
n
(number theory) Of a positive integer n, the exponent of the highest power of p that divides n. Often denoted νₚ(n). For example, ν₂(48)=4.
n
(mathematical logic) A theorem stating that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic.
n
(mathematics) Any of a set of fundamental axioms describing the natural numbers and their relationships.
n
(mathematics, number theory) An infinite sequence of integers that comprises the denominators of the closest rational approximations to the square root of 2.
adj
(mathematics) multiperfect such that σ(n) = 5n
adj
(mathematics) Of a number: equal to the sum of its proper divisors.
n
(number theory) A number that is the sum of all of its divisors except itself.
n
(mathematics) A positive integer that is an exact integer power of another positive integer.
n
(mathematics, colloquial) A numeral system that uses the golden ratio as its base.
adj
(mathematics) Based on all words in the alphabet of positive integers modulo Knuth equivalence.
n
(number theory) The conjecture that, for any positive even number n, there are infinitely many prime gaps of size n.
n
(algebra, strict sense) An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_nxⁿ+a_n-1xⁿ⁻¹+...+a_0x⁰.
n
(mathematics) The complexity class where the runtime can be bounded (from above) by a polynomial in the input size.
n
(mathematics) Any positive integer having the property that all smaller integers can be represented as sums of distinct divisors of it
n
(mathematics) A form of pullback where a function of a variable y can be rewritten as a function of x if y itself is a function of x.
n
(mathematics) The condition of being a prime number
adj
(mathematics) Having its complement closed under multiplication: said only of ideals.
n
(mathematics) An ordered set of prime numbers having a constant difference between successive elements
n
(mathematics) The factorization of an integer into prime numbers.
n
(number theory) A factor of a given integer which is also a prime number.
n
(mathematics) The factorization of a positive integer into its constituent prime numbers
n
(algebra, field theory) A field that contains no proper subfields.
n
(mathematics) difference between two successive prime numbers
n
(obsolete, number theory) Any natural number (including 1) that is divisible only by itself and 1.
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(number theory) The theorem that the number of prime numbers less than n asymptotically approaches n / ln(n) as n approaches infinity.
n
(mathematics) An ordered set of three prime numbers in which the smallest and largest of the three differ by six.
n
(number theory) Given a modulus n, a number g such that every number coprime to n is congruent (modulo n) to some power of g; equivalently, a generator of the multiplicative field of integers modulo n.
n
(mathematics, number theory) For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n.
n
(number theory) Any number belonging to the integer sequence whose nth element is the product of the first n primes.
n
(mathematics) A prime number that is one more or one less than a primorial. For example, 31 is a primorial prime, since 6# + 1 = 31.
n
(mathematics) A complex number which, when raised to the power of n, yields the radicand of its nth degree radical, and which has the greatest real part among all such numbers, and positive imaginary part in case of equality of the real parts.
n
(mathematics) The positive square root of a number; the positive number which, when squared, yields another number
adj
(mathematics) Of a number which is the product of two consecutive integers
n
(number theory) Any number of the form k·2ⁿ + 1, where k is odd, n is a positive integer, and 2ⁿ > k.
n
(mathematics) A Proth number that is prime.
adj
(mathematics) Having a nonnegative intersection product with every ample divisor and numerically eventually free if the intersection product with every positive divisor is non-negative.
adj
(mathematics) Being a factor-object of an algebraicly numerical object.
adj
Being such an integer.
n
(mathematics) A value in pseudodivision more or less corresponding to the remainder in normal division.
n
(number theory) The conjecture (disproved in 1958) that at least half of the natural numbers smaller than any given number have an odd number of prime factors.
n
(number theory) The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.
n
(number theory, modular arithmetic) For given positive integer n, any integer that is congruent to some square m² modulo n.
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(number theory) The product of the distinct prime factors of a given positive integer.
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(mathematics) A prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.
n
(combinatorics) Any one of a certain set of numbers which are guaranteed to exist by Ramsey's theorem; a positive integer which is a certain function of some given multiset of positive integers, where that "certain function" is that which yields the minimal number guaranteed to exist by Ramsey's theorem.
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The theorem that any graph labelling (with colours) of a sufficiently large complete graph contains monochromatic cliques.
n
(algebra) A theorem which states a constraint on rational solutions of a polynomial equation with integer coefficients.
n
The smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less.
n
(mathematics) A prime number belonging to one of the two groups that prime numbers are divided into. The other group is the irregular primes.
adj
(mathematics, of a number) having no factors (except the number 1) in common with a specified other number or numbers.
n
(mathematics) An iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the best in the uniform norm L_∞ sense.
n
(algebra) An algebraic structure which consists of a set with two binary operations: an additive operation and a multiplicative operation, such that the set is an abelian group under the additive operation, a monoid under the multiplicative operation, and such that the multiplicative operation is distributive with respect to the additive operation.
adj
Having to do with ring theory.
adj
Having to do with ring theory.
n
The branch of mathematics dealing with the algebraic structure of rings.
n
(number theory) A theorem stating that, given a fixed base b, the set of numbers that are the sum of a prime and a positive integer power of b has a positive lower asymptotic density.
n
(number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1.
n
(mathematics) A method for proving Gödel's incompleteness theorems without the assumption that the theory being considered is ω-consistent. While Gödel's original proof uses a sentence that states (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".
n
(mathematics) A fundamental result in Diophantine approximation to algebraic numbers, stating that these numbers cannot have many rational number approximations that are 'very good' (variously defined through history).
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(mathematics) Any of the large semiprime numbers to be factored in the RSA Factoring Challenge, a competition from 1991 to 2007 encouraging research into computational number theory.
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Any prime number of the form 2p + 1, where p is also a prime.
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(number theory) A famous open problem in mathematics, the hypothesis stating that, for every finite collection f_1,f_2,…,f_k of non-constant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds: (i) there are infinitely many positive integers n such that all of f_1(n),f_2(n),…,f_k(n) are simultaneously prime numbers, or (ii) there is an integer m>1 (called a fixed divisor) which always divides the product f_1(n)f_2(n)⋯f_k(n).
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(number theory) A technique for estimating the size of sifted sets of positive integers that satisfy a set of conditions expressed by congruences.
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(number theory) A natural number that is the product of two (not necessarily distinct) prime numbers.
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(mathematics) A conjecture concerning the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It states that for a polynomial f(z)=(z-r_1)⋯(z-r_n), qquad (n>2) with all roots r₁, ..., rₙ inside the closed unit disk |z| ≤ 1, each of the n roots is at a distance no more than 1 from at least one critical point.
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(mathematics) Used to describe prime numbers that differ from each other by six.
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(mathematics) Either of a pair of prime numbers that differ by six.
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(mathematics) A particular refinement of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.
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(number theory) An odd natural number k such that k⨯2ⁿ+1 is composite for all natural numbers n.
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(mathematics) An ancient algorithm for finding prime numbers that works by discarding multiples from a list of potential primes.
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(algebra) A root of a polynomial equation which has multiplicity one.
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(mathematics) A conjecture in combinatorial number theory, stating that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times).
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(number theory) Any of several extremely large numbers used as upper bounds for the smallest natural number x for which 𝜋(x)> operatorname li(x), where 𝜋 is the prime-counting function and li is the logarithmic integral function. These bounds have since been improved by others.
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(mathematics) A theorem in probability theory that extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
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(number theory) A function, denoted by S(n) for some positive integer n, that yields the smallest number s such that n divides the factorial s!. For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!, so S(8) = 4.
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(mathematics, of a number) That factors completely into small prime numbers.
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Any prime number p where 2p+1 is also prime.
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(set theory) A theorem that describes the largest possible families of finite sets none of which contain any other sets in the family.
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(mathematics) A positive integer that is the product of three distinct prime factors.
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(algebra, character theory) (of a character χ of a representation of a group G) A field K over which a K-representation of G exists which includes the character χ; (of a group G) a field over which a K-representation of G exists which includes every irreducible character in G.
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(mathematics) A theorem used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed.
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(mathematics) Stirling number of the first kind
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(mathematics) The number of ways to partition a set of n objects into k non-empty subsets, denoted by S(n,k).
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(mathematics) An accurate approximation for factorials.
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(mathematics) A theorem expressing the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of p.
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(number theory) A theorem that gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations.
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(mathematics, combinatorics) The result of deranging a number. In equations, it is usually symbolised by an exclamation mark (!) before the number being deranged (as opposed to factorials, where the exclamation mark occurs after the number).
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(number theory) A positive integer whose abundancy index is greater than that of any lesser positive integer.
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(mathematics) A positive integer n which there is a positive integer e such that d(n)/nᵉ ≥ d(k)/kᵉ for all k > 1, where d(n) is defined to be the divisor function of n
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(mathematics) Being or relating to a class of positive integers satisfying 𝜎²(n)=𝜎(𝜎(n))=2n,, where σ is the divisor function. They are a generalization of perfect numbers.
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(mathematics) Any of the subsequence of prime numbers that occupy prime-numbered positions within the sequence of prime numbers.
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(mathematics) A superset of sequences
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(mathematics) A result in combinatorics, stating that every set of integers with positive natural density contains a k-term arithmetic progression for every k.
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(number theory) A conjecture relating to the conductor and the discriminant of an elliptic curve, equivalent in a slightly modified form to the abc conjecture.
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(mathematical analysis) Any of a class of theorems which, for a given Abelian theorem, specifies conditions such that any series whose Abel sums converge (as stipulated by the Abelian theorem) is in fact convergent.
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(mathematics) The nth taxicab number, typically denoted Ta(n) or Taxicab(n), is the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways.
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Alternative form of Taylor series [(calculus) A power series representation of given infinitely differentiable function f whose terms are calculated from the function's arbitrary order derivatives at given reference point a; the series f(a)+(f'(a))/(1!)(x-a)+(f(a))/(2!)(x-a)²+(f'(a))/(3!)(x-a)³+⋯=∑ₙ₌₀∞(f⁽ⁿ⁾(a))/(n!)(x-a)ⁿ.]
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(mathematics) A lemma stating that every non-empty collection of finite character has a maximal element with respect to inclusion.
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The infinitely long binary sequence obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far.
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(mathematics) A prime number with at least 1000 decimal digits.
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(mathematics) Abbreviation of totally ordered set. [(set theory) A set having a specified total order.]
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(mathematics) (of a function) Defined on all possible inputs.
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(mathematics) A positive integer that is smaller than or equal to, and coprime to, another given positive integer.
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(mathematics) The number of positive integers not greater than a specified integer that are relatively prime to it.
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(mathematics) A pair of twin primes.
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(mathematics) A graphical depiction of the set of prime numbers, constructed by writing the positive integers in a square spiral and specially marking the primes.
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(algebra) A root of the polynomial x⁵ + x + a, where a is a complex number.
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(mathematics) Of a variety V over a field K: being dominated by a rational variety, so that its function field K(V) lies in a pure transcendental field of finite type.
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(mathematics, of an algebra) That contains an identity element.
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(mathematics) A positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself).
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(mathematics) In Ramsey theory, a theorem stating that, for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.
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(mathematical analysis) A theorem which states that any real-valued Lebesgue integrable function can be approached arbitrarily closely from below by an upper semicontinuous function and also from above by a lower semicontinuous function.
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(number theory) A (hypothetical) prime number p such that p² divides F_𝜋(p), where F_n is the Fibonacci sequence and 𝜋(p) is the pth Pisano period (the period length of the Fibonacci sequence reduced modulo p).
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(number theory) The problem of whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k.
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(number theory) A natural number that is abundant but not semiperfect; one for which the sum of the proper divisors (including 1 but not itself) is greater than the number, but no subset of those divisors sums to the number itself.
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(mathematics) A theorem stating that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.
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A squarefree integer with at least three prime factors which fall into the pattern p_x=ap_x-1+b, where a and b are some integer constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest.
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(game theory) A theorem about finite two-person games of perfect information in which the players move alternately and chance does not affect the decision-making process. It states that if the game cannot end in a draw, then one of the two players must have a winning strategy, i.e. force a win.
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(mathematics) A matrix whose entries are all zero.
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(set theory) A proposition of set theory stating that every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element.
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