In
mathematics, and particularly in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, ''N'' is a primary pseudoperfect number if it satisfies the
Egyptian fraction equation
:
where the sum is over only the
prime divisors of ''N''.
Properties
Equivalently, ''N'' is a primary pseudoperfect number if it satisfies
:
Except for the primary pseudoperfect number ''N'' = 2, this expression gives a representation for ''N'' as the sum of distinct divisors of ''N''. Therefore, each primary pseudoperfect number ''N'' (except ''N'' = 2) is also
pseudoperfect.
The eight known primary pseudoperfect numbers are
:
2,
6,
42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086 .
The first four of these numbers are one less than the corresponding numbers in
Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are
:2, 3, 7, 43, 1807, 3263443, 10650056950807, 11342371305542184 ...
, but then the two
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
diverge.
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 sometimes may provide solutions to
Znám's problem
In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Šte ...
, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set 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 that no number in the set can equal one plus the product of 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.
History
Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). Using computational search techniques, they proved the remarkable result that for each positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''r'' up to 8, there exists exactly one primary pseudoperfect number with precisely ''r'' (distinct) prime factors, namely, the ''r''th known primary pseudoperfect number. Those with 2 ≤ ''r'' ≤ 8, when reduced
modulo 288, form the
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
6, 42, 78, 114, 150, 186, 222, as was observed by Sondow and MacMillan (2017).
See also
*
Giuga number
References
* .
* .
* .
External links
*
*
{{Classes of natural numbers
Integer sequences
Egyptian fractions