Multiply perfect numbers of low abundancy - Research Commons

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Multiply perfect numbers of low
abundancy
A thesis
submitted in fulfilment
of the requirements for the Degree
of
Doctor of Philosophy
at the
University of Waikato
by
Qizhi Zhou
University of Waikato
2010
ii
Abstract
The purpose of this thesis is to investigate the properties of multiperfect num-
bers with low abundancy, and to include the structure, bounds, and density
of certain multiperfect numbers.
As a significant result of this thesis, an exploration of the structure of an
odd 4-perfect number has been made. An extension of Euler’s theorem on the
structure of any odd perfect number to odd 2k-perfect numbers has also been
obtained.
In order to study multiperfect numbers, it is necessary to discuss the fac-
torization of the sum of divisors, in particular for σ(qe), for prime q. This
concept is applied to investigate multiperfect numbers with a so-called flat
shape N = 2ap p . If some prime divisors of N are fixed then there are
1 m
···
finitely many flat even 3-perfect numbers. If N is a flat 4-perfect number and
the exponent of 2 is not congruent to 1 (mod 12), then the exponent is even.
If all odd prime divisors of N are Mersenne primes, where N is even, flat and
multiperfect, then N is a perfect number. In more general cases, some neces-
sary conditions for the divisibility by 3 of an even 4-perfect number N = 2ab
are obtained, where b is an odd positive integer.
Two new ideas, namely flat primes and thin primes, are introduced since
these appear often in multiperfect numbers. The relative density of flat primes
to all primes is given by 2 times Artin’s constant. An upper bound of the
number of thin primes is T(x) x . The sum of the reciprocals of the thin
(cid:28) log2x
primes is finite.
Acknowledgements
First, I am pleased to express my profound thanks to my chief supervisor,
Professor Kevin Broughan, who quite unselfishly aided and encouraged me at
many theoretical and computational junctures during my PhD thesis. I also
had to learn plenty of computer technology from him.
Next, I would like to express a special gratitude to my second supervisor,
Dr. Ron Sorli (from University of Technology, Sydney), who gave me valuable
mathematical and LaTeX advice. I hope to receive his continuous support in
future.
My acknowledgements extend to the Mathematics Department of Waikato
University, where I obtained great support for my research. I would also like
to thank the New Zealand Institute for Mathematics and its Applications
(NZIMA) for offering me a doctoral scholarship.
Furthermore, I am very appreciative of the assistance of Andrea Haines for
her grammatical suggestions, and Jackie Broughan for her English language
support.
Finally, I want to thank my family for their understanding and generous
support.
Contents
1 Summary of the Literature 1
1.1 Mathematical notations and symbols . . . . . . . . . . . . . . 1
1.2 A historical background . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Perfect numbers . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Multiperfect numbers . . . . . . . . . . . . . . . . . . . 11
1.3 An outline of this thesis . . . . . . . . . . . . . . . . . . . . . 13
1.4 Summary of the main findings . . . . . . . . . . . . . . . . . . 14
2 Restricted forms for an odd multiperfect number of abun-
dancy 4 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 The factorization of the sum of divisors 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Exponent of q modulo p . . . . . . . . . . . . . . . . . . . . . 34
3.3 The factorization theorem . . . . . . . . . . . . . . . . . . . . 39
3.4 Prime power values of σ(qe) . . . . . . . . . . . . . . . . . . . 45
4 Counting multiperfect numbers up to x 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Counting perfect numbers . . . . . . . . . . . . . . . . . . . . 54
4.3 Flat primes and thin primes . . . . . . . . . . . . . . . . . . . 64
4.3.1 Upper or lower thin numbers . . . . . . . . . . . . . . . 68
4.3.2 Upper or lower flat primes . . . . . . . . . . . . . . . . 70
4.3.3 Upper and lower thin primes . . . . . . . . . . . . . . . 81
4.3.4 Hardy-Littlewood-Bateman-Horn conjectures . . . . . . 84
4.4 Theorems on flat numbers . . . . . . . . . . . . . . . . . . . . 86
v
5 Even 3-perfect numbers of a flat shape 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Even perfect numbers of abundancy 4 110
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3 Lemmas and theorems . . . . . . . . . . . . . . . . . . . . . . 112
7 Other properties of multiply perfect numbers and unsolved
problems 126
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2 Upper bound for 5-perfect numbers of a flat shape . . . . . . . 126
7.3 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References 138
Chapter 1
Summary of the Literature
1.1 Mathematical notations and symbols
The lowercase letters p and q denote prime numbers.
The following notations are standard:
(a , ,a ) denotes thegreatest common divisor of theintegers a , ,a .
1 n 1 n
··· ···
[x] denotes the integer part of the real number x.
x denotes the fractional part of the real number x.
{ }
π(x) denotes the number of prime numbers less than or equal to x.
Z denotes the set of all integers.
N denotes the set of all positive integers.
P denotes the set of all prime numbers.
# denotes the number of elements in a set.
We use Landau’s O, o, and notation [71]:
(cid:28)
f(x) = O(g(x)) or f(x) g(x)
(cid:28)
for a range of a real x, there is a constant A such that the inequality
f(x) Ag(x)
| | ≤
2
holds over the range.
f(x) = o(g(x)) as x ,
→ ∞
means
f(x)
lim = 0
x g(x)
→∞
(g(x) = 0 for x sufficiently large.) The same meaning is used when x is
6 → ∞
replaced by x α, for any fixed α.
→
In this thesis, f(x) = O (x(cid:15)) means there is a constant C depending on
(cid:15)
(cid:15) > 0, such that
f(x) Cx(cid:15),
| | ≤
holds over the range of a real x.
A subset A of positive integers has asymptotic density d(A) = α, where
0 α 1, if the proportion of elements of A, among all positive integers from
≤ ≤
1 to n, has a limit α as n tends to infinity. That is,
A(n)
d(A) = lim | | = α,
n n
→∞
where A(n) = A 1,2, ,n .
∩{ ··· }
A function f(x) of a real variable x > 0 is written as o(x) if f(x) 0, as
x →
x . If A(x) = αx+o(x), then d(A) = α.
→ ∞ | |
A function f is said to have an upper bound C if f(x) C for all x in its
≤
domain.
Let k be a positive integer. Any collection of positive integers whose sum
is equal to k is said to form a partition of k [54]. For example,
5 = 4+1 = 3+2 = 3+1+1 = 2+2+1
= 2+1+1+1 = 1+1+1+1+1,
so that there are 7 partitions of 5.
3
The factorization of a positive integer N as a product of powers of distinct
primes p , i = 1, ,k can be given by
i
···
k
N = pαi.
i
i=1
Y
The notation v (N) = e means pe N, that is, pe N and pe+1 - N. v (N) = 0
p p
|| |
if p does not divide N.
The number of distinct prime divisors of N is denoted by
ω(N) = k
and the total number of prime divisors of N is denoted by
k
Ω(N) = α .
i
i=1
X
The sum of divisors function is denoted by
σ(N) = d.
dN
X|
For prime p and positive integer α, we have
pα+1 1
σ(pα) = 1+p+p2 + +pα = − .
··· p 1
−
It is well-known that σ is a multiplicative function. Given the prime factor-
ization of N we have
k k pαi+1 1
σ(N) = σ(pαi) = i − .
i p 1
i
i=1 i=1 −
Y Y
Given integers a, b and m with m > 0, we say that a is congruent to b modulo
m and write
a b (mod m)
≡
if m divides the difference a b.
−
Apowerfulnumber isapositiveintegermsuchthatforevery primenumber
p dividing m, p2 also divides m.