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Thus, an integer $n\geq1$ is a perfect number if For example, the numbers 6, 28, 496, 8128, 33550336$,\ldots,$ are perfect.

...l almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least 6 prime factors. If $m$ is an odd almost perfect number then $m(2m-1)$ is a [[Descartes number]].

...}.P$: it is not known if there are any odd perfect numbers. An ''[[almost perfect number]]'' $n$ similarly has the property that $\sigma(n) = 2n-1$: these in * Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation '''36''' (1981) 583–586. {{ZBL|0472.10007}}

...f divisors]]: these are also called ''Øre numbers''. The first seven such numbers are An even [[perfect number]] is a harmonic number.

A divisor $d$ of a natural number $n$ such that $d$ and $n/d$ are [[coprime numbers]], having no common factor other than 1. Equivalently, $d$ is a unitary di ...It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already k

...perfect rings are defined similarly. A left perfect ring need not be right perfect. The following properties of a ring $R$ are equivalent: 1) $R$ is a left perfect ring; 2) every set of pairwise orthogonal idempotents of $R$ is finite, and

A number which is close to being a [[perfect number]]. They are named for René Descartes who observed that the number would be an odd perfect number if only 22021 were a [[prime number]], since the [[Sum of divisors|s

...d $b$ are rational numbers, $b\ne 0$, and $d$ is an integer which is not a perfect square. A real number $\alpha$ is a quadratic irrationality if and only if

...ervals (adjacent intervals), the total of whose length is 1, is the Cantor perfect set (Cantor set; Cantor ternary set; Cantor discontinuum). ...hough the standard Cantor set has measure zero, there exists nowhere-dense perfect compacta on the unit interval with measure arbitrarily close to 1.

$#C+1 = 137 : ~/encyclopedia/old_files/data/P072/P.0702070 Perfect measure ...ions that occur in general measure theory do not arise if one restricts to perfect measures.

...is problem is closely related with the problem on the existence of perfect numbers. The Lucas test provides a very simple method to establish primality of these numbers. This test consists of the following (cf. [[#References|[a2]]]). Define $S_

set. There exists perfect sets (cf. [[Perfect set|Perfect set]]) of measure zero that are $ M $- is an integer, that is, whether a set of numbers is a $ U $-

...rly, these numbers are in fact perfect or abundant (i.e. "non-deficient") numbers. ...$b = 5$, $c = 7$, $abcd$ is abundant for any prime number $d > c$. Of the numbers $\leq 1000$, $52$ are abundant.

...ect ring]]) there is a proper class of p-test modules. If $R$ is not right perfect, then it is consistent with ZFC (cf. [[Set theory|Set theory]]; [[Zermelo a ...entified in [[#References|[a3]]]. If $R$ is right hereditary but not right perfect, then it is consistent with ZFC that there is a proper class of p-test modu

A code intended for the control of the functioning of an adder. When adding numbers represented in the binary number system, a single slip in the functioning o described around any two numbers of a code do not intersect (that is, if the minimum distance of the code is

image of the set of irrational numbers, an ${\mathcal A}$-set can be defined as a continuous image of the set of irrational numbers. A countable

The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule $a_1 = n$, $a_{k+1} = s(a_k)$ where $s( ...ers]]): aliquot cycles of length greater than 2 are also termed ''sociable numbers''.

...ots,X_n\}\}$ are generally called quasi-analytic $k$-algebras. If $k$ is a perfect field, an analytic ring is an [[Excellent ring|excellent ring]].

and is such that any perfect irreducible inverse image of the space $ X $ ...connected by a single-valued or multi-valued [[Perfect irreducible mapping|perfect irreducible mapping]] $ f: X \rightarrow Y $,

...mapping|Closed mapping]]; [[Open mapping|Open mapping]]; [[Perfect mapping|Perfect mapping]]; [[Quotient mapping|Quotient mapping]]). If <img align="absmiddle ...png" />-dimensional cube. 2) A pointwise-countable base is preserved under perfect (even under bifactorial) mappings. 3) Every closed mapping <img align="absm

...ar scheme|regular scheme]], i.e. if all its local rings are regular. For a perfect field $ k $ ...d is a non-singular algebraic variety. In the case of the field of complex numbers a non-singular algebraic variety has the structure of a complex [[Analytic

...were distinguished; the structure of perfect numbers (cf. [[Perfect number|Perfect number]]) was studied; and the solution in integers of the equation $ x ...mbers. Somewhat later, Eratosthenes discovered a method of obtaining prime numbers, which is still called the sieve of Eratosthenes (cf. [[Eratosthenes, sieve

dimensional balls of equal radius (see [[Geometry of numbers|Geometry of numbers]]). ...ng all classes of boundary forms. In particular, the Voronoi algorithm for perfect forms (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]).

be the perfect [[Cantor set|Cantor set]], $ P $ the numbers $ a _ {n} $

then there is an uncountable co-analytic set without perfect part, while if $ E $ ...firmative solution of Luzin's problem about parts of the series of natural numbers, starting from the [[Continuum hypothesis|continuum hypothesis]] or the neg

...ent (1989) one approach to finding good codes is to find curves with large numbers of rational points compared to their genus. This brings in advanced algebra

...thod for reducing the solution of an equation of degree 4 over the complex numbers (or, more generally, over any field of [[Characteristic_of_a_field|characte square brackets is a perfect square. For this the [[Quadratic equation|discriminant of the

be the [[Baire space|Baire space]] (homeomorphic to the space of irrational numbers). A set $ P \subset I ^ {m} $ ...aire property|Baire property]] and, if uncountable, contains a (non-empty) perfect subset; every $ A _ {3} $-

one may restrict attention to perfect probability spaces, i.e. spaces which occur in the general scheme, cannot occur in perfect probability

...ditionally, proofs are deemed to be non-elementary if they involve complex numbers. ...partitions, of additive representations, of the approximation by rational numbers, and of continued fractions. Quite often, the solution of such problems lea

These numbers are called the periodic correlation coefficients (in case $ a = b $, are called perfect. Perfect sequences whose entries are $ t $

...function $F$ on $[a,b]$ such that $F'=f$ almost everywhere, and if for any perfect set $P$ there exists a portion of $P$ on which $F$ is absolutely continuous ...approximate derivative]] is almost everywhere equal to $f$ and if, for any perfect set $P$, there exists a portion of $P$ on which $F$ is absolutely continuou

error-correcting perfect codes with $ e > 1 $ is a prime power. A perfect $ e $-

...mplete this part there is Luzin's theorem on subsets of the set of natural numbers, which reflects some properties of the remainder $ \beta \mathbf N \setmi "Luzin's theorem on subsets of the set of natural numbers" states that there is a family $ \{ A _ \alpha \} _ {\alpha < \omega _

of $ K _ {12 } $[[#References|[a9]]]; no larger numbers are known. ...o factors is a Hamilton cycle (cf. [[Hamiltonian tour|Hamiltonian tour]]). Perfect one-factorizations exist for many orders, and no order $ n $(

tuples of numbers is the set $ \{ {\langle a _ {1} \dots a _ {n-} 1 \rangle } : {\exists b tuples of numbers, it is possible to find a value of the parameter $ m $(

...of real functions on a topological space $X$, indexed by countable ordinal numbers, which are defined inductively iterating the operation of taking pointwise ...ich takes the values $0$ on the irrational numbers and $1$ on the rational numbers, is a classical example of function which does not belong to the first Bair

...itrary residue fields. There is a class field theory for local fields with perfect residue fields in terms of a certain fundamental group [[#References|[a1]]]

...ications (cf. [[Perfect compactification|Perfect compactification]]) . All perfect compactifications $ bX $ itself is perfect as well) and have, like $ \beta X $,

The numbers $ v,\ b,\ r _ {i} $, different numbers $ \lambda _ {1} \dots \lambda _ {m} $

...s the requirement that the measure $\operatorname P$ be "[[Perfect measure|perfect]]" (see {{Cite|GK}}). Probability distributions in function spaces are usua ...lso be obtained as a limit distribution (say, by considering sums of large numbers of random variables or some other random variables with sufficiently smooth

corresponds to the perfect differential ideal $ \{ F _{1} \dots F _{k} \} $ ...ifferential polynomials. The Ritt–Raudenbush basis theorem states that all perfect differential ideals are obtained in this way (a differential ideal $ I $

...the first to propose that a Poiseuille flow is unstable for large Reynolds numbers, and calculated four points of the neutral curve. For a quadratic profile o ..."> W.McF. Orr, "The stability or instability of the steady motions of a perfect liquid and of a viscous liquid II" ''Proc. R. Irish Acad. A'' , '''27'''

over the field of rational numbers equals the number of conjugacy classes of cyclic subgroups of the group. If is a [[Perfect field|perfect field]] there exists a splitting field for $ G $

...not have isolated points (an example is the space $\mathbf{Q}$ of rational numbers). All zero-dimensional spaces are completely regular. Zero-dimensionality i ...ero-dimensional space under a well-behaved mapping, for example, under a [[perfect mapping]] and under a continuous [[open mapping]] with compact inverse imag

...on can be written in dimensionless form by introducing three dimensionless numbers: the [[Mach number|Mach number]], the [[Reynolds number|Reynolds number]] a ...ber (cf. also [[Chapman–Enskog method|Chapman–Enskog method]]). This is in perfect agreement with the von Kármán relation. Even more, the Chapman–Enskog d

are equipotent. In this way one obtains infinitely many distinct cardinal numbers (cf. [[Cardinal number|Cardinal number]]). Cantor's theorem also implies th Any decreasing sequence of non-empty bounded closed sets of real numbers has a non-empty intersection. This generalizes to compact subsets of a metr

...s and codes with the same parameters, there is only one non-trivial binary perfect code. ...ther branches of mathematics, mainly the [[Geometry of numbers|geometry of numbers]] (cf. also [[#References|[a5]]]) and the theory of finite fields (cf. [[Fi

...nd $k=p/2$; in this case, all the nodes of $G$ are covered with edges (a [[perfect matching]]); and ...k$ component nodes. Now assign weights (or indeterminates over the complex numbers) $w_1$ and $w_2$ to each node and edge of $G$, respectively. Take the weigh

of complex numbers were in fact proved by E. Picard and, in contemporary terminology, establis is perfect or $ G $

...are often written down by arbitrarily listing their members by the natural numbers in such a way that every covering is refined by the next one in the sequenc ...rations of taking subspaces and countable products, and it is stable under perfect mappings. However, a whole series of regularities holding in the class of m

...(these include, for example, the set of perfect squares, the set of prime numbers, the set of decimal approximations of the number $ \sqrt 2 $,

the set of intersection numbers of blocks in a tight $ t $-design contains exactly $ s $ ...D valign="top">[3]</TD> <TD valign="top"> E.F. Assmus, H.F. Mattson, "Perfect codes and the Mathieu groups" ''Arch. Math. Basel'' , '''17''' (1966) pp

known as the perfect or canonical extension of $ \mathbf B $( denotes the set of real numbers) is complex but not canonical. In general, complexity of $ {\mathcal V} $

set]]) contains a [[Perfect set|perfect set]] and thus has the cardinality of the continuum. This result was also i of irrational numbers; and, as a consequence, $ {\mathcal A} $-

is a perfect field of characteristic $ p > 0 $, adic numbers $ \mathbf Z _{p} $.

...ge1$), separable Hilbert spaces, the Cantor set, the set of all irrational numbers etc. (these are separable complete metric spaces or Borel sets in such spac ...not belong to $\B$. It is a so-called [[A-set|analytic set]], and it is [[Perfect measure|universally measurable]].

...until meaningful cleartext emerges). However, there is no safety in large numbers alone. ...ch systems exist, but the problem of key-distribution is amplified because perfect secrecy is only possible if the key is at least as long as the cleartext an

...e natural numbers as well as adding any two natural numbers, since natural numbers can be regarded, for example, as words of the form $ 0 , 0 \mid , 0 \mid if a perfect natural number has been obtained, then terminate the process, otherwise add

...loping arithmetic, analysis, functional analysis and for studying cardinal numbers smaller than $ \aleph _ \omega $. and higher cardinal numbers.

numbers; he was an illustrious astronomer and geodesist, and necessary numbers of jurors and witnesses. Gauss also studied the

Here the numbers $ s $ An application of the theory of the number of representations of numbers by positive quadratic forms in an even number of variables has been given b

is a nowhere-dense perfect subset of $ \partial K $). The collection of numbers $ ( g; k _ {1} \dots k _ {n} ) $,

...e a countable subset of $ X $, and let $ p_1, p_2, \dots $ be non-negative numbers. Then the function of the additive subgroup of rational numbers (Vitali's example).

...s set. V.V. Golubev's dissertation Single-valued analytic functions with a perfect set of singular points (1916, cf. [[#References|[1]]]) was of great importa for all positive numbers $ p $,

is usually called the Newton binomial formula. The numbers $ C ( n, m) $ be a perfect square, while when $ v $

The relation (3) is known as an initial condition. The numbers $ t _ {0} $ The values of these parameters are never known with perfect accuracy, so that it is important to clarify the conditions ensuring the st

$U$ is defined over a perfect field $k$,

one can take the perfect Cantor set, and as $ \Pi _ {\aleph _ {0} } ^ {1} $ algebra; G. Bergman proved that numbers in $ ( 1 , 2 ) $

...ever-tighter package density, however, this is not achievable anymore in a perfect way. The effects of quantum theory may begin to have an influence on the ou ...ation of probabilistic algorithms. Especially, the creation of true random numbers become possible in the context of quantum Turing machines.