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2014-03-18T04:49:52Z

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2013-03-19T06:55:53Z

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[[Image:Znam-2-3-11-23-31.svg|thumb|301px|Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Therefore the product, 47058, is primary pseudoperfect.]]
In [[mathematics]], and particularly in [[number theory]], a '''primary pseudoperfect number''' is a number ''N'' that satisfies the [[Egyptian fraction]] equation
:<math>\sum_{p|N}\frac1p + \frac1N = 1,</math>
where the sum is over only the prime divisors of ''N''. Equivalently (as can be seen by multiplying this equation by ''N''),
:<math>\sum_{p|N}\frac{N}p + 1 = N.</math>
Except for the exceptional primary pseudoperfect number 2, this expression gives a representation for ''N'' as a sum of a set of distinct divisors of ''N''; therefore each such number (except 2) is [[Semiperfect number|pseudoperfect]].

Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). The first few primary pseudoperfect numbers are
:[[2 (number)|2]], [[6 (number)|6]], [[42 (number)|42]], 1806, 47058, 2214502422, 52495396602, ... {{OEIS|id=A054377}}.
The first four of these numbers are one less than the corresponding numbers in [[Sylvester's sequence]] but later numbers in Sylvester's sequence do not similarly correspond to primary pseudoperfect numbers. It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers.

The prime factors of primary pseudoperfect numbers may provide solutions to [[Znám's problem]] in which all members of the solution set are prime. For instance, the factors of the primary pseudoperfect number 47058 are the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement in Znám's problem that no number in the set can equal one plus the product of all the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has ''k'' primes in it, for each ''k'' ≤ 8, and conjectures that the same is true for larger ''k''.

If a primary pseudoperfect number ''N'' is one less than a prime number, then ''N''×(''N''+1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.

See also [[Giuga number]].

==References==
* {{citation
| last = Anne | first = Premchand
| title = Egyptian fractions and the inheritance problem
| journal = The College Mathematics Journal
| volume = 29
| issue = 4
| year = 1998
| pages = 296–300
| doi = 10.2307/2687685
| jstor = 2687685
| publisher = The College Mathematics Journal, Vol. 29, No. 4}}.

* {{citation
| last1 = Butske | first1 = William | last2 = Jaje | first2 = Lynda M. | last3 = Mayernik | first3 = Daniel R.
| title = On the equation <math>\scriptstyle\sum_{p|N}\frac{1}{p}+\frac{1}{N}=1</math>, pseudoperfect numbers, and perfectly weighted graphs
| journal = [[Mathematics of Computation]]
| volume = 69
| year = 2000
| pages = 407–420
| doi = 10.1090/S0025-5718-99-01088-1}}.

==External links==
* {{PlanetMath | urlname = PrimaryPseudoperfectNumber | title = Primary Pseudoperfect Number}}
* {{MathWorld | urlname = PrimaryPseudoperfectNumber | title = Primary Pseudoperfect Number}}

{{Classes of natural numbers}}

[[Category:Integer sequences]]
[[Category:Egyptian fractions]]

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2012-06-29T21:49:23Z

76.91.60.107: /* Original paper */

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2012-05-23T22:45:35Z

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{{refimprove|date=January 2012}}
In [[mathematical logic]], in particular in [[model theory]] and [[non-standard analysis]], an '''internal set''' is a set that is a member of a model.

The concept of internal sets is a tool in formulating the [[transfer principle]], which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the [[hyperreal number]]s. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical justification for their use. Roughly speaking, the idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to '''internal sets''' rather than to all sets (note that the term "language" is used in a loose sense in the above).

Edward Nelson's [[internal set theory]] is an axiomatic approach to non-standard analysis (see also Palmgren at [[constructive non-standard analysis]]). Conventional infinitary accounts of non-standard analysis also use the concept of internal sets.
==Internal sets in the ultrapower construction==
Relative to the [[ultrapower]] construction of the [[hyperreal number]]s as equivalence classes of sequences <math>\langle u_n\rangle</math>, an internal subset [''An''] of *R is one defined by a sequence of real sets <math>\langle A_n \rangle</math>, where a hyperreal <math>[u_n]</math> is said to belong to the set <math>[A_n]\subset \; ^*\!{\mathbb R}</math> if and only if the set of indices n such that <math>u_n \in A_n</math>, is a member of the [[ultrafilter]] used in the construction of *R.

More generally, an internal entity is a member of the natural extension of a real entity. Thus, every element of *R is internal; a subset of *R is internal if and only if it is a member of the natural extension <math>{ } ^* \mathcal{P}(\mathbb{R})</math> of the power set <math>\mathcal{P}(\mathbb{R})</math> of R; etc.
==Internal subsets of the reals==
Every internal subset of <math>\mathbb{R}</math> is necessarily ''finite'', (see Theorem 3.9.1 Goldblatt, 1998). In other words, every internal infinite subset of the hyperreals necessarily contains non-standard elements.

==See also==
*[[Standard part function]]

== References ==
*[[Robert Goldblatt|Goldblatt, Robert]]. ''Lectures on the [[hyperreal]]s''. An introduction to nonstandard analysis. [[Graduate Texts in Mathematics]], 188. Springer-Verlag, New York, 1998.

* {{citation | title=Non-standard analysis | author=Abraham Robinson | authorlink=Abraham Robinson | series=Princeton landmarks in mathematics and physics | publisher=Princeton University Press | year=1996 | isbn=978-0-691-04490-3 }}

{{Infinitesimals}}

[[Category:Non-standard analysis]]