Thus, an integer $n\geq1$ is a perfect number if
For example, the numbers 6, 28, 496, 8128, 33550336$,\ldots,$ are perfect.
3 KB (430 words) - 19:37, 21 March 2024
...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]].
2 KB (260 words) - 19:44, 17 November 2023
...}.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}}
2 KB (268 words) - 19:41, 17 November 2023
...f divisors]]: these are also called ''Øre numbers''. The first seven such numbers are
An even [[perfect number]] is a harmonic number.
2 KB (222 words) - 14:19, 12 November 2023
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
2 KB (317 words) - 19:43, 17 November 2023
...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
3 KB (491 words) - 19:59, 30 October 2016
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
1 KB (164 words) - 07:50, 4 November 2023
...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
683 bytes (103 words) - 20:31, 1 October 2016
...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.
4 KB (527 words) - 09:11, 26 March 2023
$#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.
8 KB (1,244 words) - 08:05, 6 June 2020
...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_
2 KB (344 words) - 06:44, 22 March 2024
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 $-
6 KB (865 words) - 22:26, 29 January 2020
...rly, these numbers are in fact perfect or abundant (i.e. "non-deficient") numbers.
...$b = 5$, $c = 7$, $abcd$ is abundant for any prime number $d > c$. Of the numbers $\leq 1000$, $52$ are abundant.
15 KB (2,194 words) - 05:54, 15 February 2024
...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
5 KB (732 words) - 17:46, 1 July 2020
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
8 KB (1,189 words) - 09:46, 26 March 2023
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
2 KB (327 words) - 14:21, 15 August 2023
The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule $a_1 = n$, $a_{k+1} = s(a_k)$ where $s(
...ers]]): aliquot cycles of length greater than 2 are also termed ''sociable numbers''.
2 KB (331 words) - 14:02, 12 November 2023
...ots,X_n\}\}$ are generally called quasi-analytic $k$-algebras. If $k$ is a perfect field, an analytic ring is an [[Excellent ring|excellent ring]].
2 KB (341 words) - 12:10, 25 August 2014
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 $,
10 KB (1,505 words) - 04:04, 9 May 2022
...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
15 KB (2,197 words) - 17:15, 7 February 2011
...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
3 KB (540 words) - 11:38, 12 October 2023
...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
10 KB (1,503 words) - 08:03, 6 June 2020
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]]]).
3 KB (421 words) - 22:10, 5 June 2020
be the perfect [[Cantor set|Cantor set]], $ P $
the numbers $ a _ {n} $
2 KB (262 words) - 22:15, 5 June 2020
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
5 KB (711 words) - 19:09, 11 January 2024
...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
5 KB (779 words) - 07:49, 26 January 2018
...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
2 KB (271 words) - 13:05, 17 December 2014
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} $-
10 KB (1,492 words) - 08:08, 6 June 2020
one may restrict attention to perfect probability spaces, i.e. spaces
which occur in the general scheme, cannot occur in perfect probability
4 KB (669 words) - 17:20, 11 May 2013
...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
13 KB (2,043 words) - 20:28, 13 October 2014
These numbers are called the periodic correlation coefficients (in case $ a = b $,
are called perfect. Perfect sequences whose entries are $ t $
9 KB (1,351 words) - 11:02, 26 March 2023
...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
5 KB (856 words) - 09:42, 12 October 2023
error-correcting perfect codes with $ e > 1 $
is a prime power. A perfect $ e $-
5 KB (772 words) - 11:10, 23 March 2023
...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 _
7 KB (1,075 words) - 19:29, 1 January 2021
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 $(
16 KB (2,342 words) - 09:33, 19 January 2021
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 $(
7 KB (998 words) - 19:35, 5 June 2020
...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
5 KB (746 words) - 08:32, 18 August 2013
...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]]]
4 KB (548 words) - 17:43, 20 December 2014
...ications (cf. [[Perfect compactification|Perfect compactification]]) . All perfect compactifications $ bX $
itself is perfect as well) and have, like $ \beta X $,
25 KB (3,639 words) - 09:53, 26 March 2023
The numbers $ v,\ b,\ r _ {i} $,
different numbers $ \lambda _ {1} \dots \lambda _ {m} $
9 KB (1,344 words) - 16:08, 6 February 2020
...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
8 KB (1,003 words) - 21:59, 21 November 2018
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 $
30 KB (4,468 words) - 18:44, 17 December 2019
...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'''
6 KB (863 words) - 20:36, 16 January 2024
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 $
10 KB (1,488 words) - 19:39, 5 June 2020
...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
8 KB (1,230 words) - 13:00, 3 January 2016
...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
9 KB (1,338 words) - 15:30, 1 July 2020
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
8 KB (1,230 words) - 20:30, 21 January 2021
...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
21 KB (3,203 words) - 06:33, 13 June 2022
...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
6 KB (1,005 words) - 20:14, 15 March 2023
of complex numbers were in fact proved by E. Picard and, in contemporary terminology, establis
is perfect or $ G $
8 KB (1,072 words) - 20:22, 21 December 2019