In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and particularly in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, ''N'' is a primary pseudoperfect number if it satisfies the
Egyptian fraction
An Egyptian fraction is a finite sum of distinct unit fractions, such as
\frac+\frac+\frac.
That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from eac ...
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 is the product of the previous terms, plus one. Its first few terms are
:2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 .
Sylvester's sequen ...
, 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 call ...
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 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
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
s 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''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
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
288, form the
arithmetic progression
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
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