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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 on values coprime to q. Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes. | https://en.wikipedia.org/wiki/Ramanujan's_sum |
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 {\displaystyle n} th prime number. Then for n ≥ 1 {\displaystyle n\geq 1} p n > n log n . {\displaystyle p_{n}>n\log n.} In 1999, Pierre Dusart proved a tighter lower bound: p n > n ( log n + log log n − 1 ) . {\displaystyle p_{n}>n(\log n+\log \log n-1).} | https://en.wikipedia.org/wiki/Rosser's_theorem |
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. | https://en.wikipedia.org/wiki/Selberg's_identity |
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 the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing between π ( x ) < li ( x ) {\displaystyle \pi (x)<\operatorname {li} (x)} and π ( x ) > li ( x ) {\displaystyle \pi (x)>\operatorname {li} (x)} near e 727.95133 < 1.397 × 10 316 . {\displaystyle e^{727.95133}<1.397\times 10^{316}.} It is not known whether it is the smallest crossing. | https://en.wikipedia.org/wiki/Skewes's_number |
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} . | https://en.wikipedia.org/wiki/Sophie_Germain's_theorem |
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 finite number of pairs of this type, but Størmer gave a procedure for finding them all. | https://en.wikipedia.org/wiki/Størmer's_theorem |
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, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its terms. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms. | https://en.wikipedia.org/wiki/Sylvester's_sequence |
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 "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem. | https://en.wikipedia.org/wiki/Szpiro's_conjecture |
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, of a number field to a zeta integral and study its properties. Using harmonic analysis, more precisely the Poisson summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the Hecke L-function. He also located the poles of the twisted zeta function. His work can be viewed as an elegant and powerful reformulation of a work of Erich Hecke on the proof of the functional equation of the Hecke L-function. Erich Hecke used a generalized theta series associated to an algebraic number field and a lattice in its ring of integers. | https://en.wikipedia.org/wiki/Tate's_thesis |
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. | https://en.wikipedia.org/wiki/Tijdeman's_theorem |
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. | https://en.wikipedia.org/wiki/Tunnell's_theorem |
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 congruence for polynomials in X holds: ∏ 1 ≤ k ≤ n − 1 ( X k − 1 ) ≡ n − ( X n − 1 ) / ( X − 1 ) mod ( X n − 1 ) {\displaystyle \prod _{1\leq k\leq n-1}\left(X^{k}-1\right)\equiv n-\left(X^{n}-1\right)/\left(X-1\right)\mod \left(X^{n}-1\right)} or: ∏ 1 ≤ k ≤ n − 1 ( X k − 1 ) ≡ n mod ( X n − 1 ) / ( X − 1 ) . {\displaystyle \prod _{1\leq k\leq n-1}\left(X^{k}-1\right)\equiv n\mod \left(X^{n}-1\right)/\left(X-1\right).} | https://en.wikipedia.org/wiki/Vantieghems_theorem |
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 known ones. There exist multiple variations of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas. | https://en.wikipedia.org/wiki/Vieta_jumping |
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 Ivan Matveyevich Vinogradov who proved it in the 1930s. Hardy and Littlewood had shown earlier that this result followed from the generalized Riemann hypothesis, and Vinogradov was able to remove this assumption. | https://en.wikipedia.org/wiki/Vinogradov's_theorem |
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 had been checked via brute force methods, thus only a finite number of cases to check remained before the odd Goldbach conjecture would be proven or disproven. In 2013, Harald Helfgott proved Goldbach's weak conjecture for all cases. | https://en.wikipedia.org/wiki/Vinogradov's_theorem |
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 (trivially) from Goldbach's weak conjecture. | https://en.wikipedia.org/wiki/Waring's_prime_number_conjecture |
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 proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants". | https://en.wikipedia.org/wiki/Hilbert–Waring_theorem |
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 any positive real number ε {\displaystyle \scriptstyle \varepsilon } one has ∑ x = M M + N exp ( 2 π i f ( x ) ) = O ( N 1 + ε ( t q + 1 N + t N k − 1 + q N k ) 2 1 − k ) as N → ∞ . {\displaystyle \sum _{x=M}^{M+N}\exp(2\pi if(x))=O\left(N^{1+\varepsilon }\left({t \over q}+{1 \over N}+{t \over N^{k-1}}+{q \over N^{k}}\right)^{2^{1-k}}\right){\text{ as }}N\to \infty .} This inequality will only be useful when q < N k , {\displaystyle q | https://en.wikipedia.org/wiki/Weyl's_inequality_(number_theory) |
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 considered similar problems around the same time. The initial terms of Sylvester's sequence almost solve this problem, except that the last chosen term equals one plus the product of the others, rather than being a proper divisor. Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each k ≥ 5 {\displaystyle k\geq 5} . | https://en.wikipedia.org/wiki/Znám's_problem |
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 solutions to Znám's problem using only odd numbers, and there remain several other open questions. | https://en.wikipedia.org/wiki/Znám's_problem |
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 _{a})} where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a. For example, take a = 2 and p = 7. The nonzero squares mod 7 are 1, 2, and 4, so (2|7) = 1 and (6|7) = −1. Multiplication by 2 on the nonzero numbers mod 7 has the cycle decomposition (1,2,4)(3,6,5), so the sign of this permutation is 1, which is (2|7). Multiplication by 6 on the nonzero numbers mod 7 has cycle decomposition (1,6)(2,5)(3,4), whose sign is −1, which is (6|7). | https://en.wikipedia.org/wiki/Zolotarev's_lemma |
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 a k − b k {\displaystyle a^{k}-b^{k}} for any positive integer k < n {\displaystyle k 1 {\displaystyle n>1} and n {\displaystyle n} is not equal to 6, then 2 n − 1 {\displaystyle 2^{n}-1} has a prime divisor not dividing any 2 k − 1 {\displaystyle 2^{k}-1} with k < n {\displaystyle k | https://en.wikipedia.org/wiki/Zsigmondy's_theorem |
In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend. | https://en.wikipedia.org/wiki/Behrend_sequence |
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 ) {\displaystyle b^{n-1}\equiv 1{\pmod {n}}} .for all integers b {\displaystyle b} which are relatively prime to n {\displaystyle n} . Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 (Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "F numbers" for short). | https://en.wikipedia.org/wiki/Carmichael_number |
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. | https://en.wikipedia.org/wiki/Carmichael_number |
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 puzzles, Root Extraction, where a professor in retirement at Colney Hatch postulates this as a general method for root extraction. | https://en.wikipedia.org/wiki/Dudeney_number |
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 contained within a partition's Ferrers diagram. The side-length of the Durfee square is known as the rank of the partition.The Durfee symbol consists of the two partitions represented by the points to the right or below the Durfee square. | https://en.wikipedia.org/wiki/Durfee_square |
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 Fermi–Dirac primes is the smallest number that does not divide the product of all previous elements. Srinivasa Ramanujan used the Fermi–Dirac primes to find the smallest number whose number of divisors is a given power of two. | https://en.wikipedia.org/wiki/Fermi–Dirac_prime |
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 studied in the case of quadratic polynomials. | https://en.wikipedia.org/wiki/Frobenius_pseudoprime |
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 integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. | https://en.wikipedia.org/wiki/Gaussian_integers |
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. | https://en.wikipedia.org/wiki/Gaussian_integers |
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 zeta-function. A name sometimes used for Hecke character is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.). | https://en.wikipedia.org/wiki/Algebraic_Hecke_character |
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 determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers: This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor. | https://en.wikipedia.org/wiki/Ramanujan_constant |
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).The requirement that x and y both be greater than 1 is important, since without it every positive integer would be a Leyland number of the form x1 + 1x. Also, because of the commutative property of addition, the condition x ≥ y is usually added to avoid double-covering the set of Leyland numbers (so we have 1 < y ≤ x). | https://en.wikipedia.org/wiki/Leyland_number |
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 approximated "quite closely" by sequences of rational numbers. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time. It is known that π and e are not Liouville numbers. | https://en.wikipedia.org/wiki/Irrationality_measure |
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 such that pi has dimension i and each point is contained in the closure of the next one. | https://en.wikipedia.org/wiki/Parshin_chain |
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 sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding. Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are: It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven. | https://en.wikipedia.org/wiki/Pierpont_prime |
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\not \equiv 1\mod n} . The first few Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... (sequence A063980 in the OEIS)Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers. Their infinitude has been proved several times, by Subbarao, Erdős, and Hardy & Subbarao. | https://en.wikipedia.org/wiki/Pillai_prime |
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. | https://en.wikipedia.org/wiki/Poincaré_series_(modular_form) |
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 assure convergence of such a series. | https://en.wikipedia.org/wiki/Poincaré_series_(modular_form) |
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 curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties. Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. | https://en.wikipedia.org/wiki/Shimura_variety |
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 Shimura varieties form a natural realm of examples for which equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology of a Shimura variety are more amenable to study than general automorphic forms; in particular, there is a construction attaching Galois representations to them. | https://en.wikipedia.org/wiki/Shimura_variety |
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 after the Hungarian mathematician Simon Sidon, who introduced the concept in his investigations of Fourier series. The main problem in the study of Sidon sequences, posed by Sidon, is to find the maximum number of elements that a Sidon sequence can contain, up to some bound x {\displaystyle x} . Despite a large body of research, the question has remained unsolved. | https://en.wikipedia.org/wiki/Sidon_sequence |
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 members of the following set are composite: { k ⋅ 2 n + 1: n ∈ N } . {\displaystyle \left\{\,k\cdot 2^{n}+1:n\in \mathbb {N} \,\right\}.} If the form is instead k × 2 n − 1 {\displaystyle k\times 2^{n}-1} , then k is a Riesel number. | https://en.wikipedia.org/wiki/Sierpinski_problem |
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 many times as needed. Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith: 4937775 = 3 · 5 · 5 · 65837while 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)in base 10. | https://en.wikipedia.org/wiki/Smith_number |
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, 1572863, ... (sequence A055010 in the OEIS)The 9th century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers. | https://en.wikipedia.org/wiki/321_prime |
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. Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography. | https://en.wikipedia.org/wiki/Wagstaff_prime |
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. | https://en.wikipedia.org/wiki/Wall–Sun–Sun_prime |
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, at which time both of Fermat's theorems were already well known to mathematicians.Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. As of April 2023, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS). | https://en.wikipedia.org/wiki/Generalized_Wieferich_prime |
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 − 1 ) ! | https://en.wikipedia.org/wiki/Wilson_prime |
+ 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). | https://en.wikipedia.org/wiki/Wilson_prime |
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 to the problem. If any others exist, they must be greater than 2 × 1013. | https://en.wikipedia.org/wiki/Wilson_prime |
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 includes a search for Wilson primes. Another search was coordinated at the Great Internet Mersenne Prime Search forum. | https://en.wikipedia.org/wiki/Wilson_prime |
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 theorem in the 19th century. | https://en.wikipedia.org/wiki/Wolstenholme_prime |
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 primes are 16843 and 2124679 (sequence A088164 in the OEIS). There are no other Wolstenholme primes less than 109. | https://en.wikipedia.org/wiki/Wolstenholme_prime |
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). | https://en.wikipedia.org/wiki/Woodall_number |
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 n = p n − 1 + p n + 1 2 . {\displaystyle p_{n}={{p_{n-1}+p_{n+1}} \over 2}.} For example, 53 is the sixteenth prime; the fifteenth and seventeenth primes, 47 and 59, add up to 106, and half of that is 53; thus 53 is a balanced prime. | https://en.wikipedia.org/wiki/Balanced_prime |
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} form a Cunningham chain of the first kind of length k + 1 {\displaystyle k+1} , while n + 1 , 2 n + 1 , … , 2 k n + 1 {\displaystyle n+1,2n+1,\dots ,2^{k}n+1} forms a Cunningham chain of the second kind. Each of the pairs 2 i n − 1 , 2 i n + 1 {\displaystyle 2^{i}n-1,2^{i}n+1} is a pair of twin primes. Each of the primes 2 i n − 1 {\displaystyle 2^{i}n-1} for 0 ≤ i ≤ k − 1 {\displaystyle 0\leq i\leq k-1} is a Sophie Germain prime and each of the primes 2 i n − 1 {\displaystyle 2^{i}n-1} for 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} is a safe prime. | https://en.wikipedia.org/wiki/Bi-twin_chain |
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 this (Ribenboim 1996, 6.I). The case k = 1 is Dirichlet's theorem. Two other special cases are well-known conjectures: there are infinitely many twin primes (n and 2 + n are primes), and there are infinitely many Sophie Germain primes (n and 1 + 2n are primes). Dickson's conjecture is further extended by Schinzel's hypothesis H. | https://en.wikipedia.org/wiki/Dickson's_conjecture |
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. After giving necessary and sufficient conditions that an integer cannot be represented in the form ax2 + by2 + cz2 for certain specific values of a, b and c, Ramanujan observed in a footnote: "(These) results may tempt us to suppose that there are similar simple results for the form ax2 + by2 + cz2 whatever are the values of a, b and c. It appears, however, that in most cases there are no such simple results." To substantiate this observation, Ramanujan discussed the form which is now referred to as Ramanujan's ternary quadratic form. | https://en.wikipedia.org/wiki/Ramanujan's_ternary_quadratic_form |
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)) | https://en.wikipedia.org/wiki/Hilbert_number |
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 interesting in their own right. The theorem is due to Dmitry Mirimanoff. | https://en.wikipedia.org/wiki/Mirimanoff's_congruence |
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. | https://en.wikipedia.org/wiki/Cusp_form |
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 totient function. There are infinitely many solutions to the equation for k {\displaystyle k} = 1so this value is excluded in the definition. The first few highly cototient numbers are: 2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, ... (sequence A100827 in the OEIS)Many of the highly cototient numbers are odd. | https://en.wikipedia.org/wiki/Highly_cototient_number |
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. Computations become harder, since integer factorization becomes harder as the numbers get larger. | https://en.wikipedia.org/wiki/Highly_cototient_number |
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 modulo n. The Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The following table compares the first 36 values of λ(n) (sequence A002322 in the OEIS) with Euler's totient function φ (in bold if they are different; the ns such that they are different are listed in OEIS: A033949). | https://en.wikipedia.org/wiki/Reduced_totient |
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 numbers). Heuristically, its complexity for factoring an integer n {\displaystyle n} is of the form: exp ( ( 1 + o ( 1 ) ) ( 32 9 log n ) 1 / 3 ( log log n ) 2 / 3 ) = L n {\displaystyle \exp \left(\left(1+o(1)\right)\left({\tfrac {32}{9}}\log n\right)^{1/3}\left(\log \log n\right)^{2/3}\right)=L_{n}\left} in O and L-notations. The SNFS has been used extensively by NFSNet (a volunteer distributed computing effort), NFS@Home and others to factorise numbers of the Cunningham project; for some time the records for integer factorization have been numbers factored by SNFS. | https://en.wikipedia.org/wiki/Special_number_field_sieve |
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 difference of at least one pair of prime numbers not exceeding 23: 5 − 3 = 2 7 − 3 = 4 11 − 5 = 6 11 − 3 = 8 13 − 3 = 10 17 − 5 = 12 17 − 3 = 14 19 − 3 = 16 23 − 5 = 18 23 − 3 = 20On the other hand, 149 is not a cluster prime because 140 < 146, and there is no way to write 140 as the difference of two primes that are less than or equal to 149. By convention, 2 is not considered to be a cluster prime. The first 23 odd primes (up to 89) are all cluster primes. The first few odd primes that are not cluster primes are 97, 127, 149, 191, 211, 223, 227, 229, ... OEIS: A038133It is not known if there are infinitely many cluster primes. | https://en.wikipedia.org/wiki/Cluster_prime |
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. | https://en.wikipedia.org/wiki/Compatible_system_of_ℓ-adic_representations |
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. | https://en.wikipedia.org/wiki/Congruence_of_squares |
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, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, ... (sequence A003273 in the OEIS) For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers. | https://en.wikipedia.org/wiki/Congruent_number |
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 \mathbb {Q} ^{*}/\mathbb {Q} ^{*2}} ,where Q ∗ {\displaystyle \mathbb {Q} ^{*}} is the set of nonzero rational numbers. Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent numbers. | https://en.wikipedia.org/wiki/Congruent_number |
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 {\displaystyle z} ) are three square numbers that are equally spaced apart from each other, then the spacing between them, z 2 − y 2 = y 2 − x 2 {\displaystyle z^{2}-y^{2}=y^{2}-x^{2}} , is called a congruum. The congruum problem is the problem of finding squares in arithmetic progression and their associated congrua. It can be formalized as a Diophantine equation: find integers x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} such that When this equation is satisfied, both sides of the equation equal the congruum. | https://en.wikipedia.org/wiki/Congruum |
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 congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number. | https://en.wikipedia.org/wiki/Congruum |
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)). | https://en.wikipedia.org/wiki/P-adic_cyclotomic_character |
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 investigations of the arithmetic of these fields (for prime n) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. | https://en.wikipedia.org/wiki/Cyclotomic_fields |
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 8 is deficient. Denoting by σ(n) the sum of divisors, the value 2n − σ(n) is called the number's deficiency. In terms of the aliquot sum s(n), the deficiency is n − s(n). | https://en.wikipedia.org/wiki/Deficient_number |
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 | https://en.wikipedia.org/wiki/Diophantine_quintuple |
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. | https://en.wikipedia.org/wiki/Factorion |
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. | https://en.wikipedia.org/wiki/Formula_for_primes |
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). The first frugal number which is not a prime power is 1029 = 3 × 73. In base 2, thirty-two is a frugal number, since 32 = 25 is written in base 2 as 100000 = 10101. The term economical number has been used for a frugal number, but also for a number which is either frugal or equidigital. | https://en.wikipedia.org/wiki/Frugal_number |
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 / p {\displaystyle 1/p} repeats the digits of the corresponding cyclic number infinitely, as does that of a / p {\displaystyle a/p} with rotation of the digits for any a between 1 and p − 1. The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime. That is, the multiplicative order ordp b = p − 1, which is equivalent to b being a primitive root modulo p. The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers". | https://en.wikipedia.org/wiki/Full_reptend_prime |
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 absolute Galois group of K (i.e. the characters that factor through the Galois group of the genus field of K). | https://en.wikipedia.org/wiki/Genus_character |
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 the sequence starting with 4 2 = 16 {\displaystyle 4^{2}=16} and 1 2 + 6 2 = 37 {\displaystyle 1^{2}+6^{2}=37} eventually reaches 2 2 + 0 2 = 4 {\displaystyle 2^{2}+0^{2}=4} , the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called sad or unhappy. | https://en.wikipedia.org/wiki/Happy_number |
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 Reg Allenby (a British author and senior lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" (Guy 2004:§E34). | https://en.wikipedia.org/wiki/Happy_number |
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, 91963648, 197064960, ... (sequence A159907 in the OEIS) | https://en.wikipedia.org/wiki/Hemiperfect_number |
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: | https://en.wikipedia.org/wiki/Juggler_sequence |
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 kth root of unity modulo n. See modular arithmetic for notation and terminology. The roots of unity modulo n are exactly the integers that are coprime with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n. There exist primitive roots modulo n if and only if λ ( n ) = φ ( n ) , {\displaystyle \lambda (n)=\varphi (n),} where λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are respectively the Carmichael function and Euler's totient function. A root of unity modulo n is a primitive kth root of unity modulo n for some divisor k of λ ( n ) , {\displaystyle \lambda (n),} and, conversely, there are primitive kth roots of unity modulo n if and only if k is a divisor of λ ( n ) . {\displaystyle \lambda (n).} | https://en.wikipedia.org/wiki/Root_of_unity_modulo_n |
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 article. A right-truncatable prime is a prime which remains prime when the last ("right") digit is successively removed. | https://en.wikipedia.org/wiki/Right-truncatable_prime |
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-right-truncatable prime, since 1825711, 82571, 257, and 5 are all prime. In base 10, there are exactly 4260 left-truncatable primes, 83 right-truncatable primes, and 920,720,315 left-and-right-truncatable primes. | https://en.wikipedia.org/wiki/Right-truncatable_prime |
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 numbers).The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve, "the sieve of Josephus Flavius" because of its similarity with the counting-out game in the Josephus problem. | https://en.wikipedia.org/wiki/Lucky_number |
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 denotes the n-th lucky number, and pn the n-th prime, then Ln > pn for all sufficiently large n.Because of their apparent similarities with the prime numbers, some mathematicians have suggested that some of their common properties may also be found in other sets of numbers generated by sieves of a certain unknown form, but there is little theoretical basis for this conjecture. | https://en.wikipedia.org/wiki/Lucky_number |
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 integers a and b, even when they are not coprime. | https://en.wikipedia.org/wiki/Multiplicative_functions |
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 considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions, which are additive partitions of finite sequences of positive integers, with the addition made pointwise. | https://en.wikipedia.org/wiki/Multiplicative_partition |
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. | https://en.wikipedia.org/wiki/Multiplicative_partition |
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. | https://en.wikipedia.org/wiki/Narcissistic_number |
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 i . {\displaystyle \Omega (n):=\sum a_{i}\qquad {\mbox{if}}\qquad n=\prod p_{i}^{a_{i}}.} A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. | https://en.wikipedia.org/wiki/Almost_prime |
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 ) ! | https://en.wikipedia.org/wiki/Almost_prime |
, {\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. | https://en.wikipedia.org/wiki/Almost_prime |
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 n is a totient, it can be shown that n · 2k is a totient for all natural number k. There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes p (such as 78557 and 271129, see Sierpinski number) such that all numbers of the form 2ap are nontotient, and every odd number has an even multiple which is a nontotient. | https://en.wikipedia.org/wiki/Nontotient |
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