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Artin's Conjecture On Primitive Roots
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 2025. In fact, there is no single value of ''a'' for which Artin's conjecture is proved. Formulation Let ''a'' be an integer that is not a square number and not −1. Write ''a'' = ''a''0''b''2 with ''a''0 square-free. Denote by ''S''(''a'') the set of prime numbers ''p'' such that ''a'' is a primitive root modulo ''p''. Then the conjecture states # ''S''(''a'') has a positive asymptotic density inside the set of primes. In particular, ''S''(''a'') is infinite. # Under the conditions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Hooley
Christopher Hooley (7 August 1928 – 13 December 2018) was a British mathematician and professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in 1973. He was elected a Fellow of the Royal Society in 1983. He was also a Founding Fellow of the Learned Society of Wales The Learned Society of Wales () is a national academy, learned society and Charitable organization, charity that exists to "celebrate, recognise, preserve, protect and encourage excellence in all of the scholarly disciplines", and to serve the W .... He showed that the Hasse principle holds for non-singular cubic forms in at least nine variables.C. Hooley, ''On nonary cubic forms'', Journal für die reine und angewandte Mathematik, 386, pages 32-98, (1988) References External links * 1928 births 2018 deaths 20th-century British mathematicians 21st-century British mathematicians Academics of Car ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. * Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Number (group Theory)
A cyclic number is a natural number ''n'' such that ''n'' and φ(''n'') are coprime. Here φ is Euler's totient function. An equivalent definition is that a number ''n'' is cyclic if and only if any group of order ''n'' is cyclic. Any prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ... is clearly cyclic. All cyclic numbers are square-free.For if some prime square ''p''2 divides ''n'', then from the formula for φ it is clear that ''p'' is a common divisor of ''n'' and φ(''n''). Let ''n'' = ''p''1 ''p''2 … ''p''''k'' where the ''p''''i'' are distinct primes, then φ(''n'') = (''p''1 − 1)(''p''2 − 1)...(''p''''k'' – 1). If no ''p''''i'' divides any (''p''''j'' – 1), then ''n'' and φ(''n'') have no common (prime) divisor, and ''n'' is cyclic. The first cycl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Full Reptend Prime
In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat quotient : q_p(b) = \frac (where ''p'' does not divide ''b'') gives a cyclic number. Therefore, the base ''b'' expansion of 1/p repeats the digits of the corresponding cyclic number infinitely, as does that of 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 = ''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''. Base 10 < ...
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Stephens' Constant
Stephens' constant expresses the density of certain subsets of the prime numbers. Let a and b be two multiplicatively independent integers, that is, a^m b^n \neq 1 except when both m and n equal zero. Consider the set T(a,b) of prime numbers p such that p evenly divides a^k - b for some power k. Assuming the validity of the generalized Riemann hypothesis, the density of the set T(a,b) relative to the set of all primes is a rational multiple of : C_S = \prod_p \left(1 - \frac \right) = 0.57595996889294543964316337549249669\ldots Stephens' constant is closely related to the Artin constant C_A that arises in the study of primitive roots. :C_S= \prod_ \left( C_A + \left( \right) \right) \left( \right) See also *Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard E ... * T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Heath-Brown
David Rodney "Roger" Heath-Brown is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker. Career and research In 1979 he moved to the University of Oxford, where from 1999 he held a professorship in pure mathematics. He retired in 2016. Heath-Brown is known for many striking results. He proved that there are infinitely many prime numbers of the form ''x''3 + 2''y''3. In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. He has applied Burgess's method on character sums to the ranks of elliptic curves in families. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. Heath-Brown also showed that Linnik's constant is less than or equal to 5.5. More recently, Heath-Brown is known for his pi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Riemann Hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case). Global ''L''-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent. Overview The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows. The validity of a conditional proof does not require that the CPA be true, only that ''if it were true'' it would lead to the consequent. Conditional proofs are of great importance in mathematics. Conditional proofs exist linking several otherwise unproven conjectures, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition's truth to follow from another proposition than to prove it independently. A famous network of conditional proofs is the NP-complete class of complexity theory. There is a large num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
