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Primitive Root Modulo N
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo'' if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base modulo . So is a ''primitive root modulo'' if and only if is a generator of the multiplicative group of integers modulo . Gauss defined primitive roots in Article 57 of the '' Disquisitiones Arithmeticae'' (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime . In fact, the ''Disquisitiones'' contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. Elementary example The number 3 is a primitive root modulo 7 becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, 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 use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, 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 use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted . Definition For any positive integer , define as the sum of the primitive th roots of unity. It has values in depending on the factorization of into prime factors: * if is a square-free positive integer with an even number of prime factors. * if is a square-free positive integer with an odd number of prime factors. * if has a squared prime factor. The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where is the Kronecker delta, is the Liouville function, is the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent dividing its order. Infinite examples Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer groups. Another example is the direct sum of all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, see Golod–Shafarevich theorem, and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with finite exponent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punched card ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael Function
In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that :a^m \equiv 1 \pmod holds for every integer coprime to . In algebraic terms, is the exponent of the multiplicative group of integers modulo . 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 with Euler's totient function (in bold if they are different; the s such that they are different are listed in ). Numerical examples # Carmichael's function at 5 is 4, , because for any number 0 |
Multiplicative Order
In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative order of ''a'' modulo ''n'' is the order of ''a'' in the multiplicative group of the units in the ring of the integers modulo ''n''. The order of ''a'' modulo ''n'' is sometimes written as \operatorname_n(a). Example The powers of 4 modulo 7 are as follows: : \begin 4^0 &= 1 &=0 \times 7 + 1 &\equiv 1\pmod7 \\ 4^1 &= 4 &=0 \times 7 + 4 &\equiv 4\pmod7 \\ 4^2 &= 16 &=2 \times 7 + 2 &\equiv 2\pmod7 \\ 4^3 &= 64 &=9 \times 7 + 1 &\equiv 1\pmod7 \\ 4^4 &= 256 &=36 \times 7 + 4 &\equiv 4\pmod7 \\ 4^5 &= 1024 &=146 \times 7 + 2 &\equiv 2\pmod7 \\ \vdots\end The smallest positive integer ''k'' such that 4''k'' ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3. Properties Even without knowledge that we are working in the multiplicative gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congruent to modulo ; that is :a^ \equiv 1 \pmod. 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 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 is not prime. The converse of Euler's theorem is also true: if the above congruence is true, then a and n must be coprime. The theorem is further generalized by Carmichael's theorem. The theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7^, i.e. 7^ \pmod. The integers 7 and 10 are coprime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of 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 and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity Modulo N
In mathematics, namely ring theory, a ''k''th 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 \equiv 1 \pmod . If ''k'' is the smallest such exponent for ''x'', then ''x'' is called a primitive ''k''th root of unity modulo ''n''. See modular arithmetic for notation and terminology. Software to compute modular roots is available in the python library sympy.ntheory.residue_ntheory using the function nthroot_mod. See https://docs.sympy.org/latest/modules/ntheory.html. Do not confuse this with a primitive root modulo ''n'', which is a generator of the group of units of the ring of integers modulo ''n''. The primitive roots modulo ''n'' are the primitive \varphi(n)-roots of unity modulo ''n'', where \varphi is Euler's totient function. This topic deals with roots of unity modulo ''n'' where ''n'' is at times not a prime numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In other words, if ''S'' is a subset of a group ''G'', then , the ''subgroup generated by S'', is the smallest subgroup of ''G'' containing every element of ''S'', which is equal to the intersection over all subgroups containing the elements of ''S''; equivalently, is the subgroup of all elements of ''G'' that can be expressed as the finite product of elements in ''S'' and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If ''G'' = , then we say that ''S'' ''generates'' ''G'', and the elements in ''S'' are called ''generators'' or ''group generators''. If ''S'' is the empty set, then is the trivial group , since we consider t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics .... Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, and the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains animations and three-dimensional objects. The encyclopedia has been translated from the Soviet ''Matematicheskaya entsiklopediya'' (1977) originally edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles. U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
