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In number theory, a norm group is a group of the form N L / K ( L × ) {\displaystyle N_{L/K}(L^{\times })} where L / K {\displaystyle L/K} is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm groups in K × {\displaystyle K^{\times }} are precisely the open subgroups of K × {\displaystyle K^{\times }} of finite index. | https://en.wikipedia.org/wiki/Norm_group |
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let f be a function on the natural numbers. We say that g is a normal order of f if for every ε > 0, the inequalities ( 1 − ε ) g ( n ) ≤ f ( n ) ≤ ( 1 + ε ) g ( n ) {\displaystyle (1-\varepsilon )g(n)\leq f(n)\leq (1+\varepsilon )g(n)} hold for almost all n: that is, if the proportion of n ≤ x for which this does not hold tends to 0 as x tends to infinity. It is conventional to assume that the approximating function g is continuous and monotone. | https://en.wikipedia.org/wiki/Normal_order_of_an_arithmetic_function |
In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the image lies inside the real numbers. Equivalent conditions are that F is generated over Q by one root of an integer polynomial P, all of the roots of P being real; or that the tensor product algebra of F with the real field, over Q, is isomorphic to a tensor power of R. For example, quadratic fields F of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial P irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will not be totally real, although it is a field of real numbers. | https://en.wikipedia.org/wiki/Totally_real_number_field |
The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two. Any number field that is Galois over the rationals must be either totally real or totally imaginary. | https://en.wikipedia.org/wiki/Totally_real_number_field |
In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b {\displaystyle b} that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because 3435 = 3 3 + 4 4 + 3 3 + 5 5 {\displaystyle 3435=3^{3}+4^{4}+3^{3}+5^{5}} . The term "Munchausen number" was coined by Dutch mathematician and software engineer Daan van Berkel in 2009, as this evokes the story of Baron Munchausen raising himself up by his own ponytail because each digit is raised to the power of itself. | https://en.wikipedia.org/wiki/Perfect_digit-to-digit_invariant |
In number theory, a perfect digital invariant (PDI) is a number in a given number base ( b {\displaystyle b} ) that is the sum of its own digits each raised to a given power ( p {\displaystyle p} ). | https://en.wikipedia.org/wiki/Perfect_digital_invariant |
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, σ 1 ( n ) = 2 n {\displaystyle \sigma _{1}(n)=2n} where σ 1 {\displaystyle \sigma _{1}} is the sum-of-divisors function. | https://en.wikipedia.org/wiki/Odd_perfect_number |
For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby q ( q + 1 ) / 2 {\displaystyle q(q+1)/2} is an even perfect number whenever q {\displaystyle q} is a prime of the form 2 p − 1 {\displaystyle 2^{p}-1} for positive integer p {\displaystyle p} —what is now called a Mersenne prime. | https://en.wikipedia.org/wiki/Odd_perfect_number |
Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128 (sequence A000396 in the OEIS). | https://en.wikipedia.org/wiki/Odd_perfect_number |
In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number. | https://en.wikipedia.org/wiki/Perfect_totient_number |
In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number. | https://en.wikipedia.org/wiki/Pernicious_number |
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two. Polite numbers have also been called staircase numbers because the Young diagrams which represent graphically the partitions of a polite number into consecutive integers (in the French notation of drawing these diagrams) resemble staircases. If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid.The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester, Mason, Leveque, and many other more recent authors. The polite numbers describe the possible numbers of sides of the Reinhardt polygons. | https://en.wikipedia.org/wiki/Polite_number |
In number theory, a positive integer k is said to be an Erdős–Woods number if it has the following property: there exists a positive integer a such that in the sequence (a, a + 1, …, a + k) of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, k is an Erdős–Woods number if there exists a positive integer a such that for each integer i between 0 and k, at least one of the greatest common divisors gcd(a, a + i) or gcd(a + i, a + k) is greater than 1. | https://en.wikipedia.org/wiki/Erdős–Woods_number |
In number theory, a practical number or panarithmic number is a positive integer n {\displaystyle n} such that all smaller positive integers can be represented as sums of distinct divisors of n {\displaystyle n} . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers (sequence A005153 in the OEIS) begins Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.The name "practical number" is due to Srinivasan (1948). | https://en.wikipedia.org/wiki/Practical_number |
He noted that "the subdivisions of money, weights, and measures involve numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10." His partial classification of these numbers was completed by Stewart (1954) and Sierpiński (1955). This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every even perfect number and every power of two is also a practical number. Practical numbers have also been shown to be analogous with prime numbers in many of their properties. | https://en.wikipedia.org/wiki/Practical_number |
In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple (a, b, …), the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values (n + a, n + b, …) are prime. Typically the first value in the k-tuple is 0 and the rest are distinct positive even numbers. | https://en.wikipedia.org/wiki/Prime_constellation |
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. | https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes |
One attempt by Germain to prove Fermat’s Last Theorem was to let p be a prime number of the form 8k + 7 and to let n = p – 1. In this case, x n + y n = z n {\displaystyle x^{n}+y^{n}=z^{n}} is unsolvable. Germain’s proof, however, remained unfinished. | https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes |
Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if p is an odd prime and 2p + 1 is also prime, then p must divide x, y, or z. Otherwise, x n + y n ≠ z n {\textstyle x^{n}+y^{n}\neq z^{n}} . This case where p does not divide x, y, or z is called the first case. Sophie Germain’s work was the most progress achieved on Fermat’s last theorem at that time. | https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes |
Later work by Kummer and others always divided the problem into first and second cases. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven. | https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes |
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form { p , p + 2 , p + 6 , p + 8 } . {\displaystyle \{p,\ p+2,\ p+6,\ p+8\}.} This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. | https://en.wikipedia.org/wiki/Prime_sextuplet |
In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself). | https://en.wikipedia.org/wiki/Prime_triplet |
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a that is not a multiple of n; (typically, we choose a in the range 1 < a < n − 1). | https://en.wikipedia.org/wiki/Probable_prime |
Calculate an − 1 modulo n. If the result is not 1, then n is composite. If the result is 1, then n is likely to be prime; n is then called a probable prime to base a. A weak probable prime to base a is an integer that is a probable prime to base a, but which is not a strong probable prime to base a (see below). For a fixed base a, it is unusual for a composite number to be a probable prime (that is, a pseudoprime) to that base. For example, up to 25 × 109, there are 11,408,012,595 odd composite numbers, but only 21,853 pseudoprimes base 2.: 1005 The number of odd primes in the same interval is 1,091,987,404. | https://en.wikipedia.org/wiki/Probable_prime |
In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography. Contrast with probable prime, which is likely (but not certain) to be prime, based on the output of a probabilistic primality test. In principle, every prime number can be proved to be prime in polynomial time by using the AKS primality test. Other methods which guarantee that their result is prime, but which do not work for all primes, are useful for the random generation of provable primes.Provable primes have also been generated on embedded devices. | https://en.wikipedia.org/wiki/Provable_prime |
In number theory, a pseudoprime is called an elliptic pseudoprime for (E, P), where E is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in Q ( − d ) {\displaystyle \mathbb {Q} {\big (}{\sqrt {-d}}{\big )}} , having equation y2 = x3 + ax + b with a, b integers, P being a point on E and n a natural number such that the Jacobi symbol (−d | n) = −1, if (n + 1)P ≡ 0 (mod n). The number of elliptic pseudoprimes less than X is bounded above, for large X, by X / exp ( ( 1 / 3 ) log X log log log X / log log X ) . {\displaystyle X/\exp((1/3)\log X\log \log \log X/\log \log X)\ .} | https://en.wikipedia.org/wiki/Elliptic_pseudoprime |
In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity. As an example, there are rational biquadratic and octic reciprocity laws. Define the symbol (x|p)k to be +1 if x is a k-th power modulo the prime p and -1 otherwise. Let p and q be distinct primes congruent to 1 modulo 4, such that (p|q)2 = (q|p)2 = +1. | https://en.wikipedia.org/wiki/Rational_reciprocity_law |
Let p = a2 + b2 and q = A2 + B2 with aA odd. Then ( p | q ) 4 ( q | p ) 4 = ( − 1 ) ( q − 1 ) / 4 ( a B − b A | q ) 2 . {\displaystyle (p|q)_{4}(q|p)_{4}=(-1)^{(q-1)/4}(aB-bA|q)_{2}\ .} | https://en.wikipedia.org/wiki/Rational_reciprocity_law |
If in addition p and q are congruent to 1 modulo 8, let p = c2 + 2d2 and q = C2 + 2D2. Then ( p | q ) 8 = ( q | p ) 8 = ( a B − b A | q ) 4 ( c D − d C | q ) 2 . {\displaystyle (p|q)_{8}=(q|p)_{8}=(aB-bA|q)_{4}(cD-dC|q)_{2}\ .} | https://en.wikipedia.org/wiki/Rational_reciprocity_law |
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... (sequence A007703 in the OEIS). | https://en.wikipedia.org/wiki/Regular_prime |
In number theory, a self number or Devlali number in a given number base b {\displaystyle b} is a natural number that cannot be written as the sum of any other natural number n {\displaystyle n} and the individual digits of n {\displaystyle n} . 20 is a self number (in base 10), because no such combination can be found (all n < 15 {\displaystyle n<15} give a result less than 20; all other n {\displaystyle n} give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar. | https://en.wikipedia.org/wiki/Self_prime |
In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (sequence A005835 in the OEIS) | https://en.wikipedia.org/wiki/Pseudoperfect_number |
In number theory, a sphenic number (from Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. | https://en.wikipedia.org/wiki/Sphenic_number |
In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (p1, p2, p3, ...) = (2, 3, 5, ...), pn is a strong prime if pn > pn − 1 + pn + 1/2. For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime. The first few strong primes are 11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 (sequence A051634 in the OEIS).In a twin prime pair (p, p + 2) with p > 5, p is always a strong prime, since 3 must divide p − 2, which cannot be prime. | https://en.wikipedia.org/wiki/Strong_prime |
In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28. There are only two known sublime numbers: 12 and (2126)(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1) (sequence A081357 in the OEIS). The second of these has 76 decimal digits: 6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264. == References == | https://en.wikipedia.org/wiki/Sublime_number |
In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality. Legendre symbol ( a p ) {\displaystyle \left({\frac {a}{p}}\right)} defined for p a prime, a an integer, and takes values 0, 1, or −1. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
Jacobi symbol ( a b ) {\displaystyle \left({\frac {a}{b}}\right)} defined for b a positive odd integer, a an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of b. Kronecker symbol ( a b ) {\displaystyle \left({\frac {a}{b}}\right)} defined for b any integer, a an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of b. Power residue symbol ( a b ) = ( a b ) m {\displaystyle \left({\frac {a}{b}}\right)=\left({\frac {a}{b}}\right)_{m}} is defined for a in some global field containing the mth roots of 1 ( for some m), b a fractional ideal of K built from prime ideals coprime to m. The symbol takes values in the m roots of 1. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
When m = 2 and the global field is the rationals this is more or less the same as the Jacobi symbol. Hilbert symbol The local Hilbert symbol (a,b) = is defined for a and b in some local field containing the m roots of 1 (for some m) and takes values in the m roots of 1. The power residue symbol can be written in terms of the Hilbert symbol. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
The global Hilbert symbol ( a , b ) p = ( a , b p ) = ( a , b p ) m {\displaystyle (a,b)_{p}=\left({\frac {a,b}{p}}\right)=\left({\frac {a,b}{p}}\right)_{m}} is defined for a and b in some global field K, for p a finite or infinite place of K, and is equal to the local Hilbert symbol in the completion of K at the place p. Artin symbol The local Artin symbol or norm residue symbol θ L / K ( α ) = ( α , L / K ) = ( L / K α ) {\displaystyle \theta _{L/K}(\alpha )=(\alpha ,L/K)=\left({\frac {L/K}{\alpha }}\right)} is defined for L a finite extension of the local field K, α an element of K, and takes values in the abelianization of the Galois group Gal(L/K). The global Artin symbol ψ L / K ( α ) = ( α , L / K ) = ( L / K α ) = ( ( L / K ) / α ) {\displaystyle \psi _{L/K}(\alpha )=(\alpha ,L/K)=\left({\frac {L/K}{\alpha }}\right)=((L/K)/\alpha )} is defined for α in a ray class group or idele (class) group of a global field K, and takes values in the abelianization of Gal(L/K) for L an abelian extension of K. When α is in the idele group the symbol is sometimes called a Chevalley symbol or Artin–Chevalley symbol. The local Hilbert symbol of K can be written in terms of the Artin symbol for Kummer extensions L/K, where the roots of unity can be identified with elements of the Galois group. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
The Frobenius symbol = {\displaystyle =\left} is the same as the Frobenius element of the prime P of the Galois extension L of K. "Chevalley symbol" has several slightly different meanings. It is sometimes used for the Artin symbol for ideles. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
A variation of this is the Chevalley symbol ( a , χ p ) {\displaystyle \left({\frac {a,\chi }{p}}\right)} for p a prime ideal of K, a an element of K, and χ a homomorphism of the Galois group of K to R/Z. The value of the symbol is then the value of the character χ on the usual Artin symbol. Norm residue symbol This name is for several different closely related symbols, such as the Artin symbol or the Hilbert symbol or Hasse's norm residue symbol. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
The Hasse norm residue symbol ( ( α , L / K ) / p ) = ( α , L / K p ) {\displaystyle ((\alpha ,L/K)/p)=\left({\frac {\alpha ,L/K}{p}}\right)} is defined if p is a place of K and α an element of K. It is essentially the same as the local Artin symbol for the localization of K at p. The Hilbert symbol is a special case of it in the case of Kummer extensions. Steinberg symbol (a,b). This is a generalization of the local Hilbert symbol to arbitrary fields F. The numbers a and b are elements of F, and the symbol (a,b) takes values in the second K-group of F. Galois symbol A sort of generalization of the Steinberg symbol to higher algebraic K-theory. It takes a Milnor K-group to an étale cohomology group. | https://en.wikipedia.org/wiki/Symbol_(number_theory) |
In number theory, a totative of a given positive integer n is an integer k such that 0 < k ≤ n and k is coprime to n. Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n. | https://en.wikipedia.org/wiki/Totative |
In number theory, a vampire number (or true vampire number) is a composite natural number with an even number of digits, that can be factored into two natural numbers each with half as many digits as the original number, where the two factors contain precisely all the digits of the original number, in any order, counting multiplicity. The two factors cannot both have trailing zeroes. The first vampire number is 1260 = 21 × 60. | https://en.wikipedia.org/wiki/Vampire_number |
In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. | https://en.wikipedia.org/wiki/Weird_number |
In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements. A conjecture of Paul Erdős states that this sequence contains no powers of two other than 1, 4, and 256. | https://en.wikipedia.org/wiki/Power_of_three |
In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors. That is, a number n is an Erdős–Nicolas number when there exists another number m such that ∑ d ∣ n , d ≤ m d = n . {\displaystyle \sum _{d\mid n,\ d\leq m}d=n.} The first ten Erdős–Nicolas numbers are 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800 and 91963648. (OEIS: A194472)They are named after Paul Erdős and Jean-Louis Nicolas, who wrote about them in 1975. | https://en.wikipedia.org/wiki/Erdős–Nicolas_number |
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. | https://en.wikipedia.org/wiki/Euler_product |
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. | https://en.wikipedia.org/wiki/Abundant_number |
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b: | https://en.wikipedia.org/wiki/Additive_arithmetic_function |
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 1 + 2 + 3 + 6 4 = 3 , {\displaystyle {\frac {1+2+3+6}{4}}=3,} which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer. The first numbers in the sequence of arithmetic numbers are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ... (sequence A003601 in the OEIS). | https://en.wikipedia.org/wiki/Arithmetic_number |
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. | https://en.wikipedia.org/wiki/Arithmetic_functions |
In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the polynomial. Nevertheless, certain families of integers coming from cyclotomic polynomials have factorizations given by formulas applying to the whole family, as in the examples below. | https://en.wikipedia.org/wiki/Aurifeuillian_factorization |
In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f {\displaystyle f} be an arithmetic function. We say that an average order of f {\displaystyle f} is g {\displaystyle g} if as x {\displaystyle x} tends to infinity. | https://en.wikipedia.org/wiki/Average_order_of_an_arithmetic_function |
It is conventional to choose an approximating function g {\displaystyle g} that is continuous and monotone. But even so an average order is of course not unique. In cases where the limit exists, it is said that f {\displaystyle f} has a mean value (average value) c {\displaystyle c} . | https://en.wikipedia.org/wiki/Average_order_of_an_arithmetic_function |
In number theory, an eigencurve is a rigid analytic curve that parametrizes certain p-adic families of modular forms, and an eigenvariety is a higher-dimensional generalization of this. Eigencurves were introduced by Coleman and Mazur (1998), and the term "eigenvariety" seems to have been introduced around 2001 by Kevin Buzzard (2007). | https://en.wikipedia.org/wiki/Eigencurve |
In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, including exponents but excluding exponents equal to 1. For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigital numbers (sequence A046758 in the OEIS). All prime numbers are equidigital numbers in any base. A number that is either equidigital or frugal is said to be economical. | https://en.wikipedia.org/wiki/Equidigital_number |
In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the Thue–Morse set. Non-negative integers that are not evil are called odious numbers. | https://en.wikipedia.org/wiki/Evil_numbers |
In number theory, an extravagant number (also known as a wasteful number) is a natural number in a given number base that has fewer digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 4 = 22, 6 = 2×3, 8 = 23, and 9 = 32 are extravagant numbers (sequence A046760 in the OEIS). There are infinitely many extravagant numbers in every base. | https://en.wikipedia.org/wiki/Extravagant_number |
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: x 2 ≡ q ( mod n ) . {\displaystyle x^{2}\equiv q{\pmod {n}}.} Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers. | https://en.wikipedia.org/wiki/Quadratic_residues |
In number theory, an n-Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies i m − n ≡ 1 ( mod m ) {\displaystyle i^{m-n}\equiv 1{\pmod {m}}} . The concept is named after Walter Knödel.The set of all n-Knödel numbers is denoted Kn. The special case K1 is the Carmichael numbers. There are infinitely many n-Knödel numbers for a given n. Due to Euler's theorem every composite number m is an n-Knödel number for n = m − φ ( m ) {\displaystyle n=m-\varphi (m)} where φ {\displaystyle \varphi } is Euler's totient function. | https://en.wikipedia.org/wiki/Knödel_number |
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers. | https://en.wikipedia.org/wiki/Smooth_integer |
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The n {\displaystyle n} th octahedral number O n {\displaystyle O_{n}} can be obtained by the formula: O n = n ( 2 n 2 + 1 ) 3 . {\displaystyle O_{n}={n(2n^{2}+1) \over 3}.} The first few octahedral numbers are: 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 (sequence A005900 in the OEIS). | https://en.wikipedia.org/wiki/Octahedral_number |
In number theory, an odd integer n is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base a, if a and n are coprime, and a ( n − 1 ) / 2 ≡ ( a n ) ( mod n ) {\displaystyle a^{(n-1)/2}\equiv \left({\frac {a}{n}}\right){\pmod {n}}} where ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} is the Jacobi symbol. If n is an odd composite integer that satisfies the above congruence, then n is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime) to base a. | https://en.wikipedia.org/wiki/Euler–Jacobi_pseudoprime |
In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers. In computer science, an odious number is said to have odd parity. | https://en.wikipedia.org/wiki/Odious_number |
In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than n {\displaystyle {\sqrt {n}}} . A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non- n {\displaystyle {\sqrt {n}}} -smooth. | https://en.wikipedia.org/wiki/Unusual_number |
In number theory, certain special functions have traditionally been studied, such as particular Dirichlet series and modular forms. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of the monstrous moonshine theory. | https://en.wikipedia.org/wiki/Special_functions |
In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences OEIS: A023200 and OEIS: A046132 in OEIS) below 1000 are: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487, 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971) | https://en.wikipedia.org/wiki/Cousin_prime |
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L 2 {\displaystyle L^{2}} spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups. When the group is the general linear group GL 2 {\displaystyle \operatorname {GL} _{2}} , the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform (newform) corresponds to a cuspidal representation. | https://en.wikipedia.org/wiki/Cuspidal_representation |
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple. Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers". A number that is not part of any friendly pair is called solitary. | https://en.wikipedia.org/wiki/Friendly_number |
The "abundancy" index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a "friendly number" if there exists m ≠ n such that σ(m) / m = σ(n) / n. "Abundancy" is not the same as abundance, which is defined as σ(n) − 2n. | https://en.wikipedia.org/wiki/Friendly_number |
"Abundancy" may also be expressed as σ − 1 ( n ) {\displaystyle \sigma _{-1}(n)} where σ k {\displaystyle \sigma _{k}} denotes a divisor function with σ k ( n ) {\displaystyle \sigma _{k}(n)} equal to the sum of the k-th powers of the divisors of n. The numbers 1 through 5 are all solitary. The smallest "friendly number" is 6, forming for example, the "friendly" pair 6 and 28 with "abundancy" σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. | https://en.wikipedia.org/wiki/Friendly_number |
Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers". In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function. | https://en.wikipedia.org/wiki/Friendly_number |
In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article. | https://en.wikipedia.org/wiki/Completely_multiplicative |
In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that a k ≡ 1 ( mod n ) {\textstyle a^{k}\ \equiv \ 1{\pmod {n}}} .In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n. The order of a modulo n is sometimes written as ord n ( a ) {\displaystyle \operatorname {ord} _{n}(a)} . | https://en.wikipedia.org/wiki/Multiplicative_order |
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number p, a p-adic number can be defined as a series s = ∑ i = k ∞ a i p i = a k p k + a k + 1 p k + 1 + a k + 2 p k + 2 + ⋯ {\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots } where k is an integer (possibly negative), and each a i {\displaystyle a_{i}} is a integer such that 0 ≤ a i < p . {\displaystyle 0\leq a_{i} | https://en.wikipedia.org/wiki/L-adic_integers |
In number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras. He first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. In the philosophy of mathematics, he became known for the statement that "Mathematics is the science that draws necessary conclusions". | https://en.wikipedia.org/wiki/Benjamin_Peirce |
Peirce's definition of mathematics was credited by his son, Charles Sanders Peirce, as helping to initiate the consequence-oriented philosophy of pragmatism. Like George Boole, Peirce believed that mathematics could be used to study logic. These ideas were further developed by his son Charles , who noted that logic also includes the study of faulty reasoning. In contrast, the later logicist program of Gottlob Frege and Bertrand Russell attempted to base mathematics on logic. | https://en.wikipedia.org/wiki/Benjamin_Peirce |
In number theory, integer factorization is the decomposition, of a positive integer into a product of integers. If the factors are further restricted to be prime numbers, the process is called prime factorization, and includes the test whether the given integer is prime (in this case, one has a "product" of a single factor). When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. | https://en.wikipedia.org/wiki/Factoring_integers |
The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. | https://en.wikipedia.org/wiki/Factoring_integers |
The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long.Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the integer being factored increases, the number of operations required to perform the factorization on any computer increases drastically. Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure. | https://en.wikipedia.org/wiki/Factoring_integers |
In number theory, it is often important to find factors of an integer number N. Any number N has four obvious factors: ±1 and ±N. These are called "trivial factors". Any other factor, if it exists, would be called "nontrivial". The homogeneous matrix equation A x = 0 {\displaystyle A\mathbf {x} =\mathbf {0} } , where A {\displaystyle A} is a fixed matrix, x {\displaystyle \mathbf {x} } is an unknown vector, and 0 {\displaystyle \mathbf {0} } is the zero vector, has an obvious solution x = 0 {\displaystyle \mathbf {x} =\mathbf {0} } . | https://en.wikipedia.org/wiki/Triviality_(mathematics) |
This is called the "trivial solution". Any other solutions, with x ≠ 0 {\displaystyle \mathbf {x} \neq \mathbf {0} } , are called "nontrivial". In group theory, there is a very simple group with just one element in it; this is often called the "trivial group". | https://en.wikipedia.org/wiki/Triviality_(mathematics) |
All other groups, which are more complicated, are called "nontrivial". In graph theory, the trivial graph is a graph which has only 1 vertex and no edge. Database theory has a concept called functional dependency, written X → Y {\displaystyle X\to Y} . | https://en.wikipedia.org/wiki/Triviality_(mathematics) |
The dependence X → Y {\displaystyle X\to Y} is true if Y is a subset of X, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial". It can be shown that Riemann's zeta function has zeros at the negative even numbers −2, −4, … Though the proof is comparatively easy, this result would still not normally be called trivial; however, it is in this case, for its other zeros are generally unknown and have important applications and involve open questions (such as the Riemann hypothesis). Accordingly, the negative even numbers are called the trivial zeros of the function, while any other zeros are considered to be non-trivial. | https://en.wikipedia.org/wiki/Triviality_(mathematics) |
In number theory, many arithmetic functions are integer-valued. | https://en.wikipedia.org/wiki/Integer-valued_function |
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. | https://en.wikipedia.org/wiki/Lower_numbering |
In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions. | https://en.wikipedia.org/wiki/Krasner's_lemma |
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval as n grows large. Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. | https://en.wikipedia.org/wiki/Lower_asymptotic_density |
However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of N {\displaystyle \mathbb {N} } ). If an integer is randomly selected from the interval , then the probability that it belongs to A is the ratio of the number of elements of A in to the total number of elements in . If this probability tends to some limit as n tends to infinity, then this limit is referred to as the asymptotic density of A. This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory. | https://en.wikipedia.org/wiki/Lower_asymptotic_density |
In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity. There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol ( x p ) k {\displaystyle \left({\frac {x}{p}}\right)_{k}} to be +1 if x is a k-th power modulo the prime p and -1 otherwise. Let p and q be distinct primes congruent to 1 modulo 8, such that ( p q ) 4 = ( q p ) 4 = + 1. | https://en.wikipedia.org/wiki/Octic_reciprocity |
{\displaystyle \left({\frac {p}{q}}\right)_{4}=\left({\frac {q}{p}}\right)_{4}=+1.} Let p = a2 + b2 = c2 + 2d2 and q = A2 + B2 = C2 + 2D2, with aA odd. Then ( p q ) 8 ( q p ) 8 = ( a B − b A q ) 4 ( c D − d C q ) 2 . {\displaystyle \left({\frac {p}{q}}\right)_{8}\left({\frac {q}{p}}\right)_{8}=\left({\frac {aB-bA}{q}}\right)_{4}\left({\frac {cD-dC}{q}}\right)_{2}\ .} | https://en.wikipedia.org/wiki/Octic_reciprocity |
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} . According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. | https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression |
For example, it can be used about primes in an arithmetic progression of the form a n + b {\displaystyle an+b} , where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer k ≥ 3, an AP-k (also called PAP-k) is any sequence of k primes in arithmetic progression. An AP-k can be written as k primes of the form a·n + b, for fixed integers a (called the common difference) and b, and k consecutive integer values of n. An AP-k is usually expressed with n = 0 to k − 1. This can always be achieved by defining b to be the first prime in the arithmetic progression. | https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression |
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws. | https://en.wikipedia.org/wiki/Quadratic_Gauss_sum |
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of rational integers, such as √2, and the complex number i = √−1, which generates the Gaussian integers. | https://en.wikipedia.org/wiki/Quadratic_integers |
Another common example is the non-real cubic root of unity −1 + √−3/2, which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory. | https://en.wikipedia.org/wiki/Quadratic_integers |
In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial reverse divisors are 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... (sequence A008919 in the OEIS).For instance, 1089 × 9 = 9801, the reversal of 1089, and 2178 × 4 = 8712, the reversal of 2178. The multiples produced by reversing these numbers, such as 9801 or 8712, are sometimes called palintiples. | https://en.wikipedia.org/wiki/Reverse_divisible_number |
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six: sex. If p + 2 or p + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes. | https://en.wikipedia.org/wiki/Sexy_prime |
In number theory, specifically in Diophantine approximation theory, the Markov constant M ( α ) {\displaystyle M(\alpha )} of an irrational number α {\displaystyle \alpha } is the factor for which Dirichlet's approximation theorem can be improved for α {\displaystyle \alpha } . | https://en.wikipedia.org/wiki/Markov_constant |
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