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In nuclear warfare theory, a decapitation strike is a pre-emptive first strike attack that aims to destabilize an opponent's military and civil leadership structure in the hope that it will severely degrade or destroy its capacity for nuclear retaliation. It is essentially a subset of a counterforce strike but whereas a counterforce strike seeks to destroy weapons directly, a decapitation strike is designed to remove an enemy's ability to use its weapons. Strategies against decapitation strikes include the following: Distributed command and control structures. Dispersal of political leadership and military leadership in times of tension. | https://en.wikipedia.org/wiki/Decapitation_strike |
Delegation of ICBM/SLBM launch capability to local commanders in the event of a decapitation strike. Distributed and diverse launch mechanisms.A failed decapitation strike carries the risk of immediate, massive retaliation by the targeted opponent. Many countries with nuclear weapons specifically plan to prevent decapitation strikes by employing second-strike capabilities. | https://en.wikipedia.org/wiki/Decapitation_strike |
Such countries may have mobile land-based launch, sea launch, air launch, and underground ballistic missile launch facilities so that a nuclear attack on one area of the country will not totally negate its ability to retaliate. Other nuclear warfare doctrines explicitly exclude decapitation strikes on the basis that it is better to preserve the adversary's command and control structures so that a single authority remains that is capable of negotiating a surrender or ceasefire. Implementing fail-deadly mechanisms can be a way to deter decapitation strikes and respond to successful decapitation strikes. | https://en.wikipedia.org/wiki/Decapitation_strike |
In nuclear warfare, enemy targets are divided into two types: counterforce and countervalue. A counterforce target is an element of the military infrastructure, usually either specific weapons or the bases that support them. A counterforce strike is an attack that targets those elements but leaving the civilian infrastructure, the countervalue targets, as undamaged as possible. Countervalue refers to the targeting of an opponent's cities and civilian populations. | https://en.wikipedia.org/wiki/Counterforce_strike |
An ideal counterforce attack would kill no civilians. Military attacks are prone to causing collateral damage, especially when nuclear weapons are employed. | https://en.wikipedia.org/wiki/Counterforce_strike |
In nuclear terms, many military targets are located near civilian centers, and a major counterforce strike that uses even relatively small nuclear warheads against a nation would certainly inflict many civilian casualties. Also, the requirement to use ground burst strikes to destroy hardened targets would produce far more fallout than the air bursts used to strike countervalue targets, which introduces the possibility that a counterforce strike would cause more civilian casualties over the medium term than a countervalue strike.Counterforce weapons may be seen to provide more credible deterrence in future conflict by providing options for leaders. One option considered by the Soviet Union in the 1970s was basing missiles in orbit. | https://en.wikipedia.org/wiki/Counterforce_strike |
In nuclear weapon design, the pit is the core of an implosion nuclear weapon, consisting of fissile material and any neutron reflector or tamper bonded to it. Some weapons tested during the 1950s used pits made with uranium-235 alone, or as a composite with plutonium. All-plutonium pits are the smallest in diameter and have been the standard since the early 1960s. The pit is named after the hard core found in stonefruit such as peaches and apricots. | https://en.wikipedia.org/wiki/Plutonium_core |
In nuclei, the entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted I. If the total angular momentum of a neutron is jn = ℓ + s and for a proton is jp = ℓ + s (where s for protons and neutrons happens to be 1/2 again (see note)), then the nuclear angular momentum quantum numbers I are given by: I = |jn − jp|, |jn − jp| + 1, |jn − jp| + 2, ..., (jn + jp) − 2, (jn + jp) − 1, (jn + jp)Note: The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, I, of any odd-A nucleus and integer values for any even-A nucleus. Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of hydrogen (H), carbon (C), and sodium (Na) are; The reason for the unusual fluctuations in I, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of NMR spectroscopy in organic chemistry, and MRI in nuclear medicine, due to the nuclear magnetic moment interacting with an external magnetic field. | https://en.wikipedia.org/wiki/Quantum_numbers |
In nucleic acids nanotechnology, artificial nucleic acids are designed to form molecular components that can self-assemble into stable structures for use ranging from targeted drug delivery to programmable biomaterials. DNA nanotechnology uses DNA motifs to build target shapes and arrangements. It has been used in a variety of situations, including nanorobotics, algorithmic arrays, and sensor applications. The future of DNA nanotechnology is filled with possibilities for applications.The success of DNA nanotechnology has allowed designers to develop RNA nanotechnology as a growing discipline. | https://en.wikipedia.org/wiki/RNA_origami |
RNA nanotechnology combines the simplistic design and manipulation characteristic of DNA, with the additional flexibility in structure and diversity in function similar to that of proteins. RNA's versatility in structure and function, favorable in vivo attributes, and bottom-up self-assembly is an ideal avenue for developing biomaterial and nanoparticle drug delivery. Several techniques were developed to construct these RNA nanoparticles, including RNA cubic scaffold, templated and non-templated assembly, and RNA origami. | https://en.wikipedia.org/wiki/RNA_origami |
The first work in RNA origami appeared in Science, published by Ebbe S. Andersen of Aarhus University. Researchers at Aarhus University used various 3D models and computer software to design individual RNA origami. | https://en.wikipedia.org/wiki/RNA_origami |
Once encoded as a synthetic DNA gene, adding RNA polymerase resulted in the formation of RNA origami. Observation of RNA was primarily done through atomic force microscopy, a technique that allows researchers to look at molecules a thousand times closer than would normally be possible with a conventional light microscope. | https://en.wikipedia.org/wiki/RNA_origami |
They were able to form honeycomb shapes, but determined other shapes are also possible. Cody Geary, a scholar in the field of RNA origami, described the uniqueness of the method of RNA origami. He stated that its folding recipe is encoded in the molecule itself, and determine by its sequence. The sequence gives the RNA origami both its final shape and movements of the structure as it folds. The primary challenge associated with RNA origami stems from the fact RNA folds on its own and can thus easily tangle itself. | https://en.wikipedia.org/wiki/RNA_origami |
In nucleophilic aliphatic substitution, sodium nitrite (NaNO2) replaces an alkyl halide. In the so-called Ter Meer reaction (1876) named after Edmund ter Meer, the reactant is a 1,1-halonitroalkane: The reaction mechanism is proposed in which in the first slow step a proton is abstracted from nitroalkane 1 to a carbanion 2 followed by protonation to an aci-nitro 3 and finally nucleophilic displacement of chlorine based on an experimentally observed hydrogen kinetic isotope effect of 3.3. When the same reactant is reacted with potassium hydroxide the reaction product is the 1,2-dinitro dimer. | https://en.wikipedia.org/wiki/Nitro_compounds |
In nucleophilic trifluoromethylation the active species is the CF3− anion. It was, however, widely believed that the trifluoromethyl anion is a transient species and thus cannot be isolated or observed in the condensed phase. Contrary to the popular belief, the CF3 anion, with + as a countercation, was produced and characterized by Prakash and coworkers. | https://en.wikipedia.org/wiki/Trifluoromethylation |
The challenges associated with observation of CF3 anion are alluded to its strong basic nature and its tendency to form pentacoordinated silicon species, such as − or −. The reactivity of fluoroform in combination with a strong base such as t-BuOK with carbonyl compounds in DMF is an example. Here CF3− and DMF form an hemiaminolate adduct (K). | https://en.wikipedia.org/wiki/Trifluoromethylation |
In nucleotide sugar metabolism a group of biochemicals known as nucleotide sugars act as donors for sugar residues in the glycosylation reactions that produce polysaccharides. They are substrates for glycosyltransferases. The nucleotide sugars are also intermediates in nucleotide sugar interconversions that produce some of the activated sugars needed for glycosylation reactions. Since most glycosylation takes place in the endoplasmic reticulum and golgi apparatus, there are a large family of nucleotide sugar transporters that allow nucleotide sugars to move from the cytoplasm, where they are produced, into the organelles where they are consumed.Nucleotide sugar metabolism is particularly well-studied in yeast, fungal pathogens, and bacterial pathogens, such as E. coli and Mycobacterium tuberculosis, since these molecules are required for the synthesis of glycoconjugates on the surfaces of these organisms. | https://en.wikipedia.org/wiki/Nucleotide_sugars_metabolism |
These glycoconjugates are virulence factors and components of the fungal and bacterial cell wall. These pathways are also studied in plants, but here the enzymes involved are less well understood. == References == | https://en.wikipedia.org/wiki/Nucleotide_sugars_metabolism |
In null-hypothesis significance testing, the p-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. Even though reporting p-values of statistical tests is common practice in academic publications of many quantitative fields, misinterpretation and misuse of p-values is widespread and has been a major topic in mathematics and metascience. In 2016, the American Statistician Association (ASA) made a formal statement that "p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone" and that "a p-value, or statistical significance, does not measure the size of an effect or the importance of a result" or "evidence regarding a model or hypothesis." That said, a 2019 task force by ASA has issued a statement on statistical significance and replicability, concluding with: "p-values and significance tests, when properly applied and interpreted, increase the rigor of the conclusions drawn from data." | https://en.wikipedia.org/wiki/P_value |
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers. | https://en.wikipedia.org/wiki/Ideal_number |
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers Q or a cyclotomic field Q(ζn). The latter fact and its generalizations are of fundamental importance in number theory. | https://en.wikipedia.org/wiki/Modular_curves |
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for n > 2, the Fermat curve of equation x n + y n = 1 {\displaystyle x^{n}+y^{n}=1} has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1). | https://en.wikipedia.org/wiki/Rational_point |
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture. | https://en.wikipedia.org/wiki/Tate_conjecture |
In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ρ: GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist of V, denoted V(1), is the representation on the tensor product V⊗Qp(1), where Qp(1) is the p-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure Ks of K). More generally, if m is a positive integer, the mth Tate twist of V, denoted V(m), is the tensor product of V with the m-fold tensor product of Qp(1). | https://en.wikipedia.org/wiki/Tate_twist |
Denoting by Qp(−1) the dual representation of Qp(1), the -mth Tate twist of V can be defined as V ⊗ Q p ( − 1 ) ⊗ m . {\displaystyle V\otimes \mathbf {Q} _{p}(-1)^{\otimes m}.} == References == | https://en.wikipedia.org/wiki/Tate_twist |
In number theory and combinatorics rank of a partition of a positive integer is a certain integer associated with the partition. Dyson introduced the concept in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the mathematician Srinivasa Ramanujan. | https://en.wikipedia.org/wiki/Freeman_Dyson |
In number theory and combinatorics, a multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn a partition. The concept is also found in the theory of Lie algebras. | https://en.wikipedia.org/wiki/Multipartition |
In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1The only partition of zero is the empty sum, having no parts. | https://en.wikipedia.org/wiki/Ferrers_diagram |
The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition function p(n). | https://en.wikipedia.org/wiki/Ferrers_diagram |
So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. | https://en.wikipedia.org/wiki/Ferrers_diagram |
In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of the numbers in S2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem".There is an optimization version of the partition problem, which is to partition the multiset S into two subsets S1, S2 such that the difference between the sum of elements in S1 and the sum of elements in S2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice.The partition problem is a special case of two related problems: In the subset sum problem, the goal is to find a subset of S whose sum is a certain target number T given as input (the partition problem is the special case in which T is half the sum of S). | https://en.wikipedia.org/wiki/Partition_problem |
In multiway number partitioning, there is an integer parameter k, and the goal is to decide whether S can be partitioned into k subsets of equal sum (the partition problem is the special case in which k = 2). However, it is quite different than the 3-partition problem: in that problem, the number of subsets is not fixed in advance – it should be |S|/3, where each subset must have exactly 3 elements. 3-partition is much harder than partition – it has no pseudo-polynomial time algorithm unless P = NP. | https://en.wikipedia.org/wiki/Partition_problem |
In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n {\displaystyle n} elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race). Starting from n = 0 {\displaystyle n=0} , these numbers are The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number or the faces of all dimensions of a permutohedron. | https://en.wikipedia.org/wiki/Ordered_Bell_number |
In number theory and mathematical logic, a Meertens number in a given number base b {\displaystyle b} is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam. | https://en.wikipedia.org/wiki/Meertens_number |
In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955. | https://en.wikipedia.org/wiki/Minimum_overlap_problem |
In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by M ( n ) = ∑ k = 1 n μ ( k ) {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)} for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function. See the article on the Mertens conjecture for more information about the connection between M(n) and the Riemann hypothesis. From the formula μ ( n ) = ∑ gcd ( k , n ) = 1 1 ≤ k ≤ n e 2 π i k n , {\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}e^{2\pi i{\frac {k}{n}}},} it follows that the Mertens function is given by: M ( n ) = − 1 + ∑ a ∈ F n e 2 π i a , {\displaystyle M(n)=-1+\sum _{a\in {\mathcal {F}}_{n}}e^{2\pi ia},} where Fn is the Farey sequence of order n. This formula is used in the proof of the Franel–Landau theorem. | https://en.wikipedia.org/wiki/Moebius_function |
In number theory one may study a Diophantine equation, for example, modulo p for all primes p, looking for constraints on solutions. The next step is to look modulo prime powers, and then for solutions in the p-adic field. This kind of local analysis provides conditions for solution that are necessary. In cases where local analysis (plus the condition that there are real solutions) provides also sufficient conditions, one says that the Hasse principle holds: this is the best possible situation. | https://en.wikipedia.org/wiki/Local_analysis |
It does for quadratic forms, but certainly not in general (for example for elliptic curves). The point of view that one would like to understand what extra conditions are needed has been very influential, for example for cubic forms. Some form of local analysis underlies both the standard applications of the Hardy–Littlewood circle method in analytic number theory, and the use of adele rings, making this one of the unifying principles across number theory. | https://en.wikipedia.org/wiki/Local_analysis |
In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if p B p − 1 ≡ − 1 ( mod p ) . {\displaystyle pB_{p-1}\equiv -1{\pmod {p}}.} It is named after Takashi Agoh and Giuseppe Giuga. | https://en.wikipedia.org/wiki/Agoh–Giuga_conjecture |
In number theory the n conjecture is a conjecture stated by Browkin & Brzeziński (1994) as a generalization of the abc conjecture to more than three integers. | https://en.wikipedia.org/wiki/N_conjecture |
In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in A below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive integers are in A.More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or the set of primes, if A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A. Examples: The natural density of cofinite sets of positive integers is 1, so each of them contains almost all positive integers. Almost all positive integers are composite. | https://en.wikipedia.org/wiki/Almost_all |
Almost all even positive numbers can be expressed as the sum of two primes. : 489 Almost all primes are isolated. Moreover, for every positive integer g, almost all primes have prime gaps of more than g both to their left and to their right; that is, there is no other prime between p − g and p + g. | https://en.wikipedia.org/wiki/Almost_all |
In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x and is known as Khinchin's constant. That is, for x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}}}\;} it is almost always true that lim n → ∞ ( a 1 a 2 . . . | https://en.wikipedia.org/wiki/Khinchin's_constant |
a n ) 1 / n = K 0 {\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}} where K 0 {\displaystyle K_{0}} is Khinchin's constant K 0 = ∏ r = 1 ∞ ( 1 + 1 r ( r + 2 ) ) log 2 r ≈ 2.6854520010 … {\displaystyle K_{0}=\prod _{r=1}^{\infty }{\left(1+{1 \over r(r+2)}\right)}^{\log _{2}r}\approx 2.6854520010\dots } (sequence A002210 in the OEIS)(with ∏ {\displaystyle \prod } denoting the product over all sequence terms). Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose. Among the numbers whose continued fraction expansions apparently do have this property (based on numerical evidence) are π, the Euler-Mascheroni constant γ, Apéry's constant ζ(3), and Khinchin's constant itself. However, this is unproven. Among the numbers x whose continued fraction expansions are known not to have this property are rational numbers, roots of quadratic equations (including the golden ratio Φ and the square roots of integers), and the base of the natural logarithm e. Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хинчин) in older mathematical literature. | https://en.wikipedia.org/wiki/Khinchin's_constant |
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof. The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of 2023. In fact, there is no single value of a for which Artin's conjecture is proved. | https://en.wikipedia.org/wiki/Artin_constant |
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over a field Z p {\displaystyle \mathbb {Z} _{p}} . The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. | https://en.wikipedia.org/wiki/Berlekamp–Rabin_algorithm |
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 1 {\displaystyle n>1} , there always exists at least one prime number such that n < p < 2 n . {\displaystyle n | https://en.wikipedia.org/wiki/Infinitude_of_prime_numbers |
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3 {\displaystyle n>3} , there always exists at least one prime number p {\displaystyle p} with n < p < 2 n − 2. {\displaystyle n 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that n < p < 2 n . {\displaystyle n | https://en.wikipedia.org/wiki/Bertrand's_postulate |
In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if p1, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then p n # = p 1 ⋯ p n > p n + 1 2 . {\displaystyle p_{n}\#=p_{1}\cdots p_{n}>p_{n+1}^{2}.} (the middle product is short-hand for the primorial p n # {\displaystyle p_{n}\#} of pn) Mathematician Denis Hanson showed an upper bound where n # ≤ 3 n {\displaystyle n\#\leq 3^{n}} . | https://en.wikipedia.org/wiki/Bonse's_inequality |
In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. | https://en.wikipedia.org/wiki/Brocard's_conjecture |
However, it remains unproven as of 2022. The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... OEIS: A050216. Legendre's conjecture that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for pn ≥ 3 since pn+1 − pn ≥ 2. | https://en.wikipedia.org/wiki/Brocard's_conjecture |
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 (sequence A065421 in the OEIS). Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. | https://en.wikipedia.org/wiki/Brun's_constant |
In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (x + i)2, i = 1, 2, ..., M,... for some integer x. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: Does there exist a positive integer M such that, for all integers x and a, the quantity (x + n)2 + a cannot be a square for more than M consecutive values of n, unless a = 0? | https://en.wikipedia.org/wiki/Büchi's_problem |
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P, Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U12(1, −1) = 144 and its equivalent U12(−1, −1) = −144. In particular, for n greater than 12, the nth Fibonacci number F(n) has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof. | https://en.wikipedia.org/wiki/Carmichael's_theorem |
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853. | https://en.wikipedia.org/wiki/Chebyshev's_bias |
In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes. | https://en.wikipedia.org/wiki/Chen's_theorem |
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that p n + 1 − p n = O ( ( log p n ) 2 ) , {\displaystyle p_{n+1}-p_{n}=O((\log p_{n})^{2}),\ } where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his heuristic actually supports the stronger statement lim sup n → ∞ p n + 1 − p n ( log p n ) 2 = 1 , {\displaystyle \limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{(\log p_{n})^{2}}}=1,} and sometimes this formulation is called Cramér's conjecture. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture. Neither form has yet been proven or disproven. | https://en.wikipedia.org/wiki/Cramér_conjecture |
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers α {\displaystyle \alpha } and N {\displaystyle N} , with 1 ≤ N {\displaystyle 1\leq N} , there exist integers p {\displaystyle p} and q {\displaystyle q} such that 1 ≤ q ≤ N {\displaystyle 1\leq q\leq N} and | q α − p | ≤ 1 ⌊ N ⌋ + 1 < 1 N . {\displaystyle \left|q\alpha -p\right|\leq {\frac {1}{\lfloor N\rfloor +1}}<{\frac {1}{N}}.} Here ⌊ N ⌋ {\displaystyle \lfloor N\rfloor } represents the integer part of N {\displaystyle N} . | https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_diophantine_approximation |
This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality 0 < | α − p q | < 1 q 2 {\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}} is satisfied by infinitely many integers p and q. This shows that any irrational number has irrationality measure at least 2. This corollary also shows that the Thue–Siegel–Roth theorem, a result in the other direction, provides essentially the tightest possible bound, in the sense that the bound on rational approximation of algebraic numbers cannot be improved by increasing the exponent beyond 2. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the golden ratio ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} can be much more easily verified to be inapproximable beyond exponent 2. This exponent is referred to as the irrationality measure. | https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_diophantine_approximation |
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression a , a + d , a + 2 d , a + 3 d , … , {\displaystyle a,\ a+d,\ a+2d,\ a+3d,\ \dots ,\ } and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo d containing a's coprime to d. | https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions |
In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method. Unlike for other factor base methods, its run-time bound comes with a rigorous proof that does not rely on conjectures about the smoothness properties of the values taken by a polynomial. The algorithm was designed by John D. Dixon, a mathematician at Carleton University, and was published in 1981. | https://en.wikipedia.org/wiki/Dixon's_factorization_method |
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 ( mod p ) if there is an integer x such that x 2 ≡ a ( mod p ) , − 1 ( mod p ) if there is no such integer. {\displaystyle a^{\tfrac {p-1}{2}}\equiv {\begin{cases}\;\;\,1{\pmod {p}}&{\text{ if there is an integer }}x{\text{ such that }}x^{2}\equiv a{\pmod {p}},\\-1{\pmod {p}}&{\text{ if there is no such integer. | https://en.wikipedia.org/wiki/Euler's_criterion |
}}\end{cases}}} Euler's criterion can be concisely reformulated using the Legendre symbol: ( a p ) ≡ a p − 1 2 ( mod p ) . {\displaystyle \left({\frac {a}{p}}\right)\equiv a^{\tfrac {p-1}{2}}{\pmod {p}}.} The criterion first appeared in a 1748 paper by Leonhard Euler. | https://en.wikipedia.org/wiki/Euler's_criterion |
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function, then a raised to the power φ ( n ) {\displaystyle \varphi (n)} is congruent to 1 modulo n; that is a φ ( n ) ≡ 1 ( mod n ) . {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.} In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime.The converse of Euler's theorem is also true: if the above congruence is true, then a {\displaystyle a} and n {\displaystyle n} must be coprime. | https://en.wikipedia.org/wiki/Euler's_theorem |
The theorem is further generalized by Carmichael's theorem. The theorem may be used to easily reduce large powers modulo n {\displaystyle n} . For example, consider finding the ones place decimal digit of 7 222 {\displaystyle 7^{222}} , i.e. 7 222 ( mod 10 ) {\displaystyle 7^{222}{\pmod {10}}} . | https://en.wikipedia.org/wiki/Euler's_theorem |
The integers 7 and 10 are coprime, and φ ( 10 ) = 4 {\displaystyle \varphi (10)=4} . So Euler's theorem yields 7 4 ≡ 1 ( mod 10 ) {\displaystyle 7^{4}\equiv 1{\pmod {10}}} , and we get 7 222 ≡ 7 4 × 55 + 2 ≡ ( 7 4 ) 55 × 7 2 ≡ 1 55 × 7 2 ≡ 49 ≡ 9 ( mod 10 ) {\displaystyle 7^{222}\equiv 7^{4\times 55+2}\equiv (7^{4})^{55}\times 7^{2}\equiv 1^{55}\times 7^{2}\equiv 49\equiv 9{\pmod {10}}} . In general, when reducing a power of a {\displaystyle a} modulo n {\displaystyle n} (where a {\displaystyle a} and n {\displaystyle n} are coprime), one needs to work modulo φ ( n ) {\displaystyle \varphi (n)} in the exponent of a {\displaystyle a}: if x ≡ y ( mod φ ( n ) ) {\displaystyle x\equiv y{\pmod {\varphi (n)}}} , then a x ≡ a y ( mod n ) {\displaystyle a^{x}\equiv a^{y}{\pmod {n}}} .Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with n being a product of two large prime numbers, and the security of the system is based on the difficulty of factoring such an integer. | https://en.wikipedia.org/wiki/Euler's_theorem |
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ ( n ) {\displaystyle \varphi (n)} or ϕ ( n ) {\displaystyle \phi (n)} , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. | https://en.wikipedia.org/wiki/Euler_totient |
Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1. | https://en.wikipedia.org/wiki/Euler_totient |
Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } ). It is also used for defining the RSA encryption system. | https://en.wikipedia.org/wiki/Euler_totient |
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} can satisfy the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} for any integer value of n {\displaystyle n} greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica, where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century, and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics, and prior to its proof it was in the Guinness Book of World Records for "most difficult mathematical problems". | https://en.wikipedia.org/wiki/Mathematical_conjecture |
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. | https://en.wikipedia.org/wiki/Fermat’s_Last_Theorem |
Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. | https://en.wikipedia.org/wiki/Fermat’s_Last_Theorem |
It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th and 20th centuries. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs. | https://en.wikipedia.org/wiki/Fermat’s_Last_Theorem |
In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 by James Whitbread Lee Glaisher, it states that the number of partitions of an integer n {\displaystyle n} into parts not divisible by d {\displaystyle d} is equal to the number of partitions in which no part is repeated d {\displaystyle d} or more times. This generalizes a result established in 1748 by Leonhard Euler for the case d = 2 {\displaystyle d=2} . | https://en.wikipedia.org/wiki/Glaisher's_theorem |
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum. )This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, then this would also be true. | https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture |
For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3). In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture. As of 2018, the proof is widely accepted in the mathematics community, but it has not yet been published in a peer-reviewed journal. | https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture |
The proof was accepted for publication in the Annals of Mathematics Studies series in 2015, and has been undergoing further review and revision since; fully-refereed chapters in close to final form are being made public in the process.Some state the conjecture as Every odd number greater than 7 can be expressed as the sum of three odd primes.This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture. | https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture |
In number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128. | https://en.wikipedia.org/wiki/Grimm's_conjecture |
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory. | https://en.wikipedia.org/wiki/Hilbert's_irreducibility_theorem |
In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that The condition that ξ is irrational cannot be omitted. Moreover the constant 5 {\displaystyle {\sqrt {5}}} is the best possible; if we replace 5 {\displaystyle {\sqrt {5}}} by any number A > 5 {\displaystyle A>{\sqrt {5}}} and we let ξ = ( 1 + 5 ) / 2 {\displaystyle \xi =(1+{\sqrt {5}})/2} (the golden ratio) then there exist only finitely many relatively prime integers m, n such that the formula above holds. The theorem is equivalent to the claim that the Markov constant of every number is larger than 5 {\displaystyle {\sqrt {5}}} . | https://en.wikipedia.org/wiki/Hurwitz's_theorem_(number_theory) |
In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values { p ( n ) ≠ 0: n ∈ N } {\displaystyle {\begin{Bmatrix}p(n)\neq 0:n\in \mathbb {N} \end{Bmatrix}}} , then the set of primes that divide some member of S is infinite. | https://en.wikipedia.org/wiki/Schur's_theorem |
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives. | https://en.wikipedia.org/wiki/Iwasawa_theory |
In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in base 10: 9981 – 1899 = 8082 8820 – 0288 = 8532 8532 – 2358 = 6174 7641 – 1467 = 61746174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations. The algorithm runs on any natural number in any given number base. | https://en.wikipedia.org/wiki/Kaprekar's_routine |
In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if p is a prime number, x ∈ Z / p Z {\displaystyle x\in \mathbb {Z} /p\mathbb {Z} } , and f ( x ) ∈ Z {\displaystyle \textstyle f(x)\in \mathbb {Z} } is a polynomial with integer coefficients, then either: every coefficient of f(x) is divisible by p, or f(x) ≡ 0 (mod p) has at most deg f(x) solutionswhere deg f(x) is the degree of f(x). If the modulus is not prime, then it is possible for there to be more than deg f(x) solutions. | https://en.wikipedia.org/wiki/Lagrange's_theorem_(number_theory) |
In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. | https://en.wikipedia.org/wiki/Lemoine's_conjecture |
In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by Gustav Lochs.The theorem states that for almost all real numbers in the interval (0,1), the number of terms m of the number's continued fraction expansion that are required to determine the first n places of the number's decimal expansion behaves asymptotically as follows: lim n → ∞ m n = 6 ln ( 2 ) ln ( 10 ) π 2 ≈ 0.97027014 {\displaystyle \lim _{n\to \infty }{\frac {m}{n}}={\frac {6\ln(2)\ln(10)}{\pi ^{2}}}\approx 0.97027014} (sequence A086819 in the OEIS).As this limit is only slightly smaller than 1, this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the representation by approximately one decimal place. The decimal system is the last positional system for which each digit carries less information than one continued fraction quotient; going to base-11 (changing ln ( 10 ) {\displaystyle \ln(10)} to ln ( 11 ) {\displaystyle \ln(11)} in the equation) makes the above value exceed 1. The reciprocal of this limit, π 2 6 ln ( 2 ) ln ( 10 ) ≈ 1.03064083 {\displaystyle {\frac {\pi ^{2}}{6\ln(2)\ln(10)}}\approx 1.03064083} (sequence A062542 in the OEIS),is twice the base-10 logarithm of Lévy's constant. | https://en.wikipedia.org/wiki/Lochs'_theorem |
A prominent example of a number not exhibiting this behavior is the golden ratio—sometimes known as the "most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit. == References == | https://en.wikipedia.org/wiki/Lochs'_theorem |
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas. | https://en.wikipedia.org/wiki/Lucas'_theorem |
In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the prime-counting function and λ is greater than 1 then π ( x + ( log x ) λ ) − π ( x ) ( log x ) λ − 1 {\displaystyle {\frac {\pi (x+(\log x)^{\lambda })-\pi (x)}{(\log x)^{\lambda -1}}}} does not have a limit as x tends to infinity; more precisely the limit superior is greater than 1, and the limit inferior is less than 1. The Cramér model of primes predicts incorrectly that it has limit 1 when λ≥2 (using the Borel–Cantelli lemma). | https://en.wikipedia.org/wiki/Maier's_theorem |
In number theory, Mazur's control theorem, introduced by Mazur (1972), describes the behavior in Zp extensions of the Selmer group of an abelian variety over a number field. | https://en.wikipedia.org/wiki/Mazur's_control_theorem |
In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation Q(x) = 0has a non-zero real solution, then it has a non-zero rational solution (the converse is obvious). By clearing the denominators, an integral solution x may also be found. | https://en.wikipedia.org/wiki/Meyer's_theorem |
Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement: A rational quadratic form in five or more variables represents zero over the field Qp of the p-adic numbers for all p.Meyer's theorem is best possible with respect to the number of variables: there are indefinite rational quadratic forms Q in four variables which do not represent zero. One family of examples is given by Q(x1,x2,x3,x4) = x12 + x22 − p(x32 + x42),where p is a prime number that is congruent to 3 modulo 4. This can be proved by the method of infinite descent using the fact that if the sum of two perfect squares is divisible by such a p then each summand is divisible by p. | https://en.wikipedia.org/wiki/Meyer's_theorem |
In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function ⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor } is a prime number for all positive natural numbers n. This constant is named after William Harold Mills who proved in 1947 the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unproven, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... (sequence A051021 in the OEIS). | https://en.wikipedia.org/wiki/Mills'_constant |
In number theory, Moessner's theorem or Moessner's magic is related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers 1 n , 2 n , 3 n , 4 n , ⋯ , {\displaystyle 1^{n},2^{n},3^{n},4^{n},\cdots ~,} with n ≥ 1 , {\displaystyle n\geq 1~,} by recursively manipulating the sequence of integers algebraically. The algorithm was first published by Alfred Moessner in 1951; the first proof of its validity was given by Oskar Perron that same year.For example, for n = 2 {\displaystyle n=2} , one can remove every even number, resulting in ( 1 , 3 , 5 , 7 ⋯ ) {\displaystyle (1,3,5,7\cdots )} , and then add each odd number to the sum of all previous elements, providing ( 1 , 4 , 9 , 16 , ⋯ ) = ( 1 2 , 2 2 , 3 2 , 4 2 ⋯ ) {\displaystyle (1,4,9,16,\cdots )=(1^{2},2^{2},3^{2},4^{2}\cdots )} . | https://en.wikipedia.org/wiki/Moessner's_theorem |
In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and H(n) = the largest exponent appearing in the unique prime factorization of a natural number n > 1, then Niven's constant is given by lim n → ∞ 1 n ∑ j = 1 n H ( j ) = 1 + ∑ k = 2 ∞ ( 1 − 1 ζ ( k ) ) = 1.705211 … {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}H(j)=1+\sum _{k=2}^{\infty }\left(1-{\frac {1}{\zeta (k)}}\right)=1.705211\dots } where ζ is the Riemann zeta function.In the same paper Niven also proved that ∑ j = 1 n h ( j ) = n + c n + o ( n ) {\displaystyle \sum _{j=1}^{n}h(j)=n+c{\sqrt {n}}+o({\sqrt {n}})} where h(1) = 1, h(n) = the smallest exponent appearing in the unique prime factorization of each natural number n > 1, o is little o notation, and the constant c is given by c = ζ ( 3 2 ) ζ ( 3 ) , {\displaystyle c={\frac {\zeta ({\frac {3}{2}})}{\zeta (3)}},} and consequently that lim n → ∞ 1 n ∑ j = 1 n h ( j ) = 1. {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}h(j)=1.} | https://en.wikipedia.org/wiki/Niven's_constant |
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. | https://en.wikipedia.org/wiki/Ostrowski's_theorem |
In number theory, Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians . Put q = e i π τ {\displaystyle q=e^{i\pi \tau }} , for τ {\displaystyle \tau } a complex number in the upper half plane, and define the theta function: The relation between θ ( − 1 / τ ) {\displaystyle \theta (-1/\tau )} and θ ( τ ) {\displaystyle \theta (\tau )} turns out to be important for number theory, since this kind of relation is one of the defining properties of a modular form. By choosing s ( x ) = e − π x 2 {\displaystyle s(x)=e^{-\pi x^{2}}} and using the fact that S ( f ) = e − π f 2 , {\displaystyle S(f)=e^{-\pi f^{2}},} one can conclude: by putting 1 / λ = τ / i . {\displaystyle {1/\lambda }={\sqrt {\tau /i}}.} It follows from this that θ 8 {\displaystyle \theta ^{8}} has a simple transformation property under τ ↦ − 1 / τ {\displaystyle \tau \mapsto {-1/\tau }} and this can be used to prove Jacobi's formula for the number of different ways to express an integer as the sum of eight perfect squares. | https://en.wikipedia.org/wiki/Poisson_summation_formula |
In number theory, Proth's theorem is a primality test for Proth numbers. It states that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which a p − 1 2 ≡ − 1 ( mod p ) , {\displaystyle a^{\frac {p-1}{2}}\equiv -1{\pmod {p}},} then p is prime. In this case p is called a Proth prime. This is a practical test because if p is prime, any chosen a has about a 50 percent chance of working, furthermore, since the calculation is mod p, only values of a smaller than p have to be taken into consideration. | https://en.wikipedia.org/wiki/Proth's_theorem |
In practice, however, a quadratic nonresidue of p is found via a modified Euclid's algorithm and taken as the value of a, since if a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. For such an a the Legendre symbol is ( a p ) = − 1. {\displaystyle \left({\frac {a}{p}}\right)=-1.} | https://en.wikipedia.org/wiki/Proth's_theorem |
Thus, in contrast to many Monte Carlo primality tests (randomized algorithms that can return a false positive), the primality testing algorithm based on Proth's theorem is a Las Vegas algorithm, always returning the correct answer but with a running time that varies randomly. Note that if a is chosen to be a quadratic nonresidue as described above, the runtime is constant, safe for the time spent on finding such a quadratic nonresidue. Finding such a value is very fast compared to the actual test. | https://en.wikipedia.org/wiki/Proth's_theorem |
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