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c_ltix5mm1j53s | In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},} where (a, q) = 1 means that a only takes ... | Ramanujan's sum |
c_o8z4224ua7sq | In number theory, Rosser's theorem states that the n {\displaystyle n} th prime number is greater than n log n {\displaystyle n\log n} , where log {\displaystyle \log } is the natural logarithm function. It was published by J. Barkley Rosser in 1939.Its full statement is: Let p n {\displaystyle p_{n}} be the n {\disp... | Rosser's theorem |
c_uadbyrcqs55c | In number theory, Selberg's identity is an approximate identity involving logarithms of primes named after Atle Selberg. The identity, discovered jointly by Selberg and Paul Erdős, was used in the first elementary proof for the prime number theorem. | Selberg's identity |
c_xc1xrek1q7qj | In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x {\displaystyle x} for which π ( x ) > li ( x ) , {\displaystyle \pi (x)>\operatorname {li} (x),} where π is the prime-counting function and li is t... | Skewes's number |
c_rswxf3nob4m9 | In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x p + y p = z p {\displaystyle x^{p}+y^{p}=z^{p}} of Fermat's Last Theorem for odd prime p {\displaystyle p} . | Sophie Germain's theorem |
c_g3xcgmy5eym4 | In number theory, Størmer's theorem, named after Carl Størmer, 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. It follows from the Thue–Siegel–Roth theorem that there are only a... | Størmer's theorem |
c_uf4go50q5skv | In number theory, Sylvester's sequence is an integer sequence in which each term is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).Sylvester's sequence is named after James Joseph... | Sylvester's sequence |
c_a57no47tpfb8 | In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "th... | Szpiro's conjecture |
c_z9kvqhtmbsy2 | In number theory, Tate's thesis is the 1950 PhD thesis of John Tate (1950) completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, o... | Tate's thesis |
c_v85ahf4iby7j | In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation y m = x n + 1 , {\displaystyle y^{m}=x^{n}+1,} for exponents n and m greater than one, is finite. | Tijdeman's theorem |
c_oy9mb0i30zi1 | In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution. | Tunnell's theorem |
c_k1ldp4dyzwyb | In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n≥3 is prime if and only if ∏ 1 ≤ k ≤ n − 1 ( 2 k − 1 ) ≡ n mod ( 2 n − 1 ) . {\displaystyle \prod _{1\leq k\leq n-1}\left(2^{k}-1\right)\equiv n\mod \left(2^{n}-1\right).} Similarly, n is prime, if and only if the following... | Vantieghems theorem |
c_24j15r7pi1fo | In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it can be used to produce new solutions of a quadratic Diophantine equation from k... | Vieta jumping |
c_6zayrpmq0wf5 | In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after I... | Vinogradov's theorem |
c_r28zksf4un42 | The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes. The notion of "sufficiently large" was ill-defined in Vinogradov's original work, but in 2002 it was shown that 101346 is sufficiently large. Additionally numbers up to 1020 ha... | Vinogradov's theorem |
c_98vc8wedtjgp | In number theory, Waring's prime number conjecture is a conjecture related to Vinogradov's theorem, named after the English mathematician Edward Waring. It states that every odd number exceeding 3 is either a prime number or the sum of three prime numbers. It follows from the generalized Riemann hypothesis, and (trivia... | Waring's prime number conjecture |
c_79o0xeiyeps7 | In number theory, Waring's problem asks 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. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was p... | Hilbert–Waring theorem |
c_dm80oqf3sewa | In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies | c − a / q | ≤ t q − 2 , {\displaystyle |c-a/q|\leq tq^{-2},} for some t greater than or equal to 1, then for... | Weyl's inequality (number theory) |
c_p4gf7nlkyf4n | In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had conside... | Znám's problem |
c_oehc1rk4m4qs | Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed k {\displaystyle k} . It is unknown whether there are any solutio... | Znám's problem |
c_8q76s532daqh | In number theory, Zolotarev's lemma states that the Legendre symbol ( a p ) {\displaystyle \left({\frac {a}{p}}\right)} for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation: ( a p ) = ε ( π a ) {\displaystyle \left({\frac {a}{p}}\right)=\varepsilon (\pi ... | Zolotarev's lemma |
c_elth76bhqtw8 | In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 {\displaystyle a>b>0} are coprime integers, then for any integer n ≥ 1 {\displaystyle n\geq 1} , there is a prime number p (called a primitive prime divisor) that divides a n − b n {\displaystyle a^{n}-b^{n}} and does not divide... | Zsigmondy's theorem |
c_l1wc92hjp9z4 | In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend. | Behrend sequence |
c_iewh2qbhrzho | In number theory, a Carmichael number is a composite number n {\displaystyle n} , which in modular arithmetic satisfies the congruence relation: b n ≡ b ( mod n ) {\displaystyle b^{n}\equiv b{\pmod {n}}} for all integers b {\displaystyle b} . The relation may also be expressed in the form: b n − 1 ≡ 1 ( mod n ) {\displ... | Carmichael number |
c_ftjmacndsvv1 | They are infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality.The Carmichael numbers form the subset K1 of the Knödel numbers. | Carmichael number |
c_fxioyk2rqocg | In number theory, a Dudeney number in a given number base b {\displaystyle b} is a natural number equal to the perfect cube of another natural number such that the digit sum of the first natural number is equal to the second. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his pu... | Dudeney number |
c_9xj963tmbf5k | In number theory, a Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least s parts with values ≥ s. An equivalent, but more visual, definition is that the Durfee square is the largest square that is conta... | Durfee square |
c_hvve7xlrr7bw | In number theory, a Fermi–Dirac prime is a prime power whose exponent is a power of two. These numbers are named from an analogy to Fermi–Dirac statistics in physics based on the fact that each integer has a unique representation as a product of Fermi–Dirac primes without repetition. Each element of the sequence of Fer... | Fermi–Dirac prime |
c_lyurpwyrmxrc | In number theory, a Frobenius pseudoprime is a pseudoprime, whose definition was inspired by the quadratic Frobenius test described by Jon Grantham in a 1998 preprint and published in 2000. Frobenius pseudoprimes can be defined with respect to polynomials of degree at least 2, but they have been most extensively studie... | Frobenius pseudoprime |
c_ruzrlb27c6gv | In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z {\displaystyle \mathbf {Z} } or Z . {\displaystyle \mathbb {Z} .} Gaussian integ... | Gaussian integers |
c_0r7p4270qdi3 | However, Gaussian integers do not have a total ordering that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers. Gaussian integers are named after the German mathematician Carl Friedrich Gauss. | Gaussian integers |
c_sapq319939ew | In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zet... | Algebraic Hecke character |
c_kcof15n04j9g | In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field Q {\displaystyle \mathbb {Q} \left} has class number 1. Equivalently, the ring of algebraic integers of Q {\displaystyle \mathbb {Q} \left} has unique factorization.The determin... | Ramanujan constant |
c_4swvhoxsur1z | In number theory, a Leyland number is a number of the form x y + y x {\displaystyle x^{y}+y^{x}} where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 (sequence A076980 in the OEIS... | Leyland number |
c_br4arpavp473 | In number theory, a Liouville number is a real number x {\displaystyle x} with the property that, for every positive integer n {\displaystyle n} , there exists a pair of integers ( p , q ) {\displaystyle (p,q)} with q > 1 {\displaystyle q>1} such that Liouville numbers are "almost rational", and can thus be approximate... | Irrationality measure |
c_stkvadjdlr25 | In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by Parshin (1978) in order to define an analogue of the idele class group for 2-dimensional schemes. A Parshin chain of dimension s on a scheme is a finite sequence of points p0, p1, ..., ps ... | Parshin chain |
c_x1objzt8zhlc | In number theory, a Pierpont prime is a prime number of the form for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic section... | Pierpont prime |
c_lzim4jt5uk10 | In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, n ! ≡ − 1 mod p {\displaystyle n!\equiv -1\mod p} but p ≢ 1 mod n {\displaystyle p\... | Pillai prime |
c_xd74bl0syh8w | In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables. In particular, they generalize classical Eisenstein series. They are named after Henri Poincaré. | Poincaré series (modular form) |
c_owaq26jdyah2 | If Γ is a finite group acting on a domain D and H(z) is any meromorphic function on D, then one obtains an automorphic function by averaging over Γ: ∑ γ ∈ Γ H ( γ ( z ) ) . {\displaystyle \sum _{\gamma \in \Gamma }H(\gamma (z)).} However, if Γ is a discrete group, then additional factors must be introduced in order to ... | Poincaré series (modular form) |
c_xnfr9gbjm16s | In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura ... | Shimura variety |
c_l37uhou9d6z0 | Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models defined over a number field, the reflex field of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic framework for the work of Shimura. In 1979, Robert Langlands remarked that Sh... | Shimura variety |
c_xqbnniwya897 | In number theory, a Sidon sequence is a sequence A = { a 0 , a 1 , a 2 , … } {\displaystyle A=\{a_{0},a_{1},a_{2},\dots \}} of natural numbers in which all pairwise sums a i + a j {\displaystyle a_{i}+a_{j}} (for i ≤ j {\displaystyle i\leq j} ) are different. Sidon sequences are also called Sidon sets; they are named a... | Sidon sequence |
c_dikwdmh4cnbs | In number theory, a Sierpiński number is an odd natural number k such that k × 2 n + 1 {\displaystyle k\times 2^{n}+1} is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all memb... | Sierpinski problem |
c_nlqi8nzuwjan | In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as m... | Smith number |
c_hhuq6ufxi9ix | In number theory, a Thabit number, Thâbit ibn Qurra number, or 321 number is an integer of the form 3 ⋅ 2 n − 1 {\displaystyle 3\cdot 2^{n}-1} for a non-negative integer n. The first few Thabit numbers are: 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 157286... | 321 prime |
c_b1ahgwezzyno | In number theory, a Wagstaff prime is a prime number of the form 2 p + 1 3 {\displaystyle {{2^{p}+1} \over 3}} where p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr. ; the prime pages credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference. Wagstaf... | Wagstaff prime |
c_zghcb7hyrqw9 | In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. | Wall–Sun–Sun prime |
c_lumusykh8a03 | In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem,... | Generalized Wieferich prime |
c_uiu5ywccn5hm | In number theory, a Wilson prime is a prime number p {\displaystyle p} such that p 2 {\displaystyle p^{2}} divides ( p − 1 ) ! + 1 {\displaystyle (p-1)!+1} , where " ! {\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle p} divides ( p − ... | Wilson prime |
c_0b3a833spm4v | + 1 {\displaystyle (p-1)!+1} . Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham.The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). | Wilson prime |
c_yra9ktb5cogt | Costa et al. write that "the case p = 5 {\displaystyle p=5} is trivial", and credit the observation that 13 is a Wilson prime to Mathews (1892). Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied ... | Wilson prime |
c_ccccznspniju | It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval {\displaystyle } is about log log x y {\displaystyle \log \log _{x}y} .Several computer searches have been done in the hope of finding new Wilson primes. The Ibercivis distributed computing project... | Wilson prime |
c_3sfx9yfhir4e | In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this... | Wolstenholme prime |
c_iukkfsc4rsb1 | Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two. The only two known Wolstenholme ... | Wolstenholme prime |
c_x6fz57pnwpjc | In number theory, a Woodall number (Wn) is any natural number of the form W n = n ⋅ 2 n − 1 {\displaystyle W_{n}=n\cdot 2^{n}-1} for some natural number n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS). | Woodall number |
c_sxww82bi3ds3 | In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number p n {\displaystyle p_{n}} , where n is its index in the ordered set of prime numbers, p ... | Balanced prime |
c_1xzxsrus8fco | In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers n − 1 , n + 1 , 2 n − 1 , 2 n + 1 , … , 2 k n − 1 , 2 k n + 1 {\displaystyle n-1,n+1,2n-1,2n+1,\dots ,2^{k}n-1,2^{k}n+1\,} in which every number is prime.The numbers n − 1 , 2 n − 1 , … , 2 k n − 1 {\displaystyle n-1,2n-1,\dots ,2^{k}n-1... | Bi-twin chain |
c_co4rx7xcfy6p | In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by Dickson (1904) 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 th... | Dickson's conjecture |
c_qlf8gelaz9la | In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x2 + y2 + 10z2 with integral values for x, y and z. Srinivasa Ramanujan considered this expression in a footnote in a paper published in 1916 and briefly discussed the representability of integers in this form. Aft... | Ramanujan's ternary quadratic form |
c_e9aj2yhdrr8r | In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... (sequence A016813 in the OEIS)) | Hilbert number |
c_bxwfu9rkclnb | In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are i... | Mirimanoff's congruence |
c_cmnosryd1bug | In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. | Cusp form |
c_lp5d476p352x | In number theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation x − ϕ ( x ) = k {\displaystyle x-\phi (x)=k} than any other integer below k {\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totie... | Highly cototient number |
c_o6fd2qtljcds | In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 29 modulo 30.The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Com... | Highly cototient number |
c_kc0friznz4lr | In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} holds for every integer a coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers... | Reduced totient |
c_t8iz38by3cg0 | In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form re ± s, where r and s are small (for instance Mersenne nu... | Special number field sieve |
c_6yhgf7afhkr3 | In number theory, a cluster prime is a prime number p such that every even positive integer k ≤ p − 3 can be written as the difference between two prime numbers not exceeding p (OEIS: A038134). For example, the number 23 is a cluster prime because 23 − 3 = 20, and every even integer from 2 to 20, inclusive, is the diff... | Cluster prime |
c_btfdv95pbzgc | In number theory, a compatible system of ℓ-adic representations is an abstraction of certain important families of ℓ-adic Galois representations, indexed by prime numbers ℓ, that have compatibility properties for almost all ℓ. | Compatible system of ℓ-adic representations |
c_jtidd87i4fzn | In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. | Congruence of squares |
c_pii9n7pdbooo | In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.The sequence of (integer) congruent numbers starts with 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 3... | Congruent number |
c_8mnk305owkrm | If q is a congruent number then s2q is also a congruent number for any natural number s (just by multiplying each side of the triangle by s), and vice versa. This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group Q ∗ / Q ∗ 2 {\displaystyle \... | Congruent number |
c_d9t0fwh8irhv | In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x 2 {\displaystyle x^{2}} , y 2 {\displaystyle y^{2}} , and z 2 {\displaystyle z^{2}} (for integers x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyl... | Congruum |
c_5lnno59waoxj | Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle. Congrua are also closely connected with congruent numbers: every congruu... | Congruum |
c_s2brf7mg5rw3 | In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ: G → AutR(R(1)) ≈ GL(1, R)). | P-adic cyclotomic character |
c_6oyvvk5zcutv | In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep... | Cyclotomic fields |
c_s0qq1c8hbk1e | In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so ... | Deficient number |
c_78iug6yd1ucw | In number theory, a diophantine m-tuple is a set of m positive integers { a 1 , a 2 , a 3 , a 4 , … , a m } {\displaystyle \{a_{1},a_{2},a_{3},a_{4},\ldots ,a_{m}\}} such that a i a j + 1 {\displaystyle a_{i}a_{j}+1} is a perfect square for any 1 ≤ i < j ≤ m . {\displaystyle 1\leq i | Diophantine quintuple |
c_jk1k53tt2yco | In number theory, a factorion in a given number base b {\displaystyle b} is a natural number that equals the sum of the factorials of its digits. The name factorion was coined by the author Clifford A. Pickover. | Factorion |
c_o08axi4zu5ae | In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. | Formula for primes |
c_1csbtwkdc5na | In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers (sequence A046759 in the OEIS). Th... | Frugal number |
c_p5o39esv9tjg | In number theory, a full reptend prime, full repetend prime, proper prime: 166 or long prime in base b is an odd prime number p such that the Fermat quotient q p ( b ) = b p − 1 − 1 p {\displaystyle q_{p}(b)={\frac {b^{p-1}-1}{p}}} (where p does not divide b) gives a cyclic number. Therefore, the base b expansion of 1 ... | Full reptend prime |
c_g57qb8vge2zu | In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a real character of the narrow class group of K. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified real characters of the abs... | Genus character |
c_tkid9d40g3mm | In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1 2 + 3 2 = 10 {\displaystyle 1^{2}+3^{2}=10} , and 1 2 + 0 2 = 1 {\displaystyle 1^{2}+0^{2}=1} . On the other hand, 4 is not a happy number because th... | Happy number |
c_3d0zezq2h7w5 | More generally, a b {\displaystyle b} -happy number is a natural number in a given number base b {\displaystyle b} that eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} .The origin of happy numbers is not clear. Happy numbers were brought to the attention of R... | Happy number |
c_16lb2p3km3pm | In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the divisor function, the sum of all positive divisors of n. The first few hemiperfect numbers are: 2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91... | Hemiperfect number |
c_038jiqjt3kl9 | In number theory, a juggler sequence is an integer sequence that starts with a positive integer a0, with each subsequent term in the sequence defined by the recurrence relation: | Juggler sequence |
c_qx8l4hx50of8 | In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) x k ≡ 1 ( mod n ) {\displaystyle x^{k}\equiv 1{\pmod {n}}} . If k is the smallest such exponent for x, then x is called a primitive kt... | Root of unity modulo n |
c_sadv79m29uk4 | In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this... | Right-truncatable prime |
c_swquautcynui | 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime. A left-and-right-truncatable prime is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one- or two-digit prime. 1825711 is an example of a left-and-ri... | Right-truncatable prime |
c_18f875ou0cov | In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural n... | Lucky number |
c_caic55mykv5g | Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and twin primes also appear to occur with similar frequency. However, if Ln... | Lucky number |
c_09l6k7ew4uzo | In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever a and b are coprime. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive int... | Multiplicative functions |
c_e1nfngja6pds | In number theory, a multiplicative partition or unordered factorization of an integer n {\displaystyle n} is a way of writing n {\displaystyle n} as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number n {\displaystyle n} is itself cons... | Multiplicative partition |
c_vsg1n8sx0d4b | Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been introduced by Hughes & Shallit (1983). The Latin name "factorisatio numerorum" had been used previously. MathWorld uses the term unordered factorization. | Multiplicative partition |
c_9r9h65tboqwq | In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base b {\displaystyle b} is a number that is the sum of its own digits each raised to the power of the number of digits. | Narcissistic number |
c_r5hkbri4qfek | In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents): Ω ( n ) := ∑ a i if n = ∏ p i a... | Almost prime |
c_ktvfcutnjukr | The set of k-almost primes is usually denoted by Pk. The smallest k-almost prime is 2k. The first few k-almost primes are: The number πk(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to: π k ( n ) ∼ ( n log n ) ( log log n ) k − 1 ( k − 1 ) ... | Almost prime |
c_j3ik1v15cm64 | , {\displaystyle \pi _{k}(n)\sim \left({\frac {n}{\log n}}\right){\frac {(\log \log n)^{k-1}}{(k-1)! }},} a result of Landau. See also the Hardy–Ramanujan theorem. | Almost prime |
c_tfv8v4ag2miy | An even nontotient may be one more than a prime number, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(p) = p − 1. Also, a pronic number n(n − 1) is certainly not a nontotient if n is prime since φ(p2) = p(p − 1). If a natural number... | Nontotient |
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