Exploration on odd perfect numbers
Shatin Pui Ying College
(Registration Number : 51082)
Teacher
Mr. Lai Chi Keung
Team Members
Man Kong Fard,
Yip Shun,
Wong Tsz Ching,
Li Ling Chit,
Leung Ka Wai
A report submitted to the Scientific Committee of the Hang Lung Mathematics Award, 2006
19 Aug 2006
Abstract
The aim of our project is to investigate the existence of odd perfect numbers. In our investigation, we obtained three main results.
First of all, we analyzed the properties of hypothetical odd perfect numbers and showed that any odd perfect number must exist in the form of , where , and is prime . Using this result, we further studied the unit digits of and M2 and succeeded to rule out some of their possibilities.
Secondly, we explored the properties of the function , which represents the sum of all factors of n. We found out that if , and b is a positive integer other than 1, then . By this property, we proved that multiples of a perfect number would never be a perfect number.
After all, we made use of the above property of to further our study on multiply perfect numbers. Surprisingly, we had found that 2 times of an odd perfect number is a 3-perfect number and concluded that if the number of 3-perfect number is finite, then the number of odd perfect number is also finite.
First Stage
First of all, let’s define what a perfect number is. By the definition in [3], a natural number Ω is said to be perfect if and only if the sumof all the proper factors of Ω is equal to itself, i.e.. Obviously, when A and B are relatively prime,
.
Let , where are distinct prime numbers,
then it is well-known that
…… (*)
We now focus on finding the general form of an odd perfect number and its properties.
Theorem 1:
If Ω is an odd perfect number, then for some satisfying both and is prime.
Proof:
Let Ω=, where are distinct prime numbers.
As Ω is odd, 2Ω is a multiple of 2 but not a multiple of 4.
By (*), we obtained the following equation:
WLOG, we assume that is the only even factor, then
are odd numbers. Consequently, are all even and is odd.
Now is either or
If. then
= ,
which is a multiple of 4.
This violates with the fact that 2Ω is a multiple of 2 but not a multiple of 4.
So must be in a form of .
Similarly, is either in the form or .
If, then
Since N is an integer, so is a multiple of 4 which violates with the fact that 2Ω is a multiple of 2 but not a multiple of 4.
So = and .
As are all even numbers, is a perfect square.
And so we can conclude that , where
Clearly and is prime.
(Q.E.D.)
Throughout this paper, we denote an odd perfect number by
,
where and
We now explore on the properties of the unit digits of and using our Main Theorem 1.
The unit digit of can be 1, 3, 5, 7 or 9, and the unit digit of can be 1, 5 or 9. So there are totally 15 combinations that are shown in the following table.
Unit digits / / / / // 1, 1 / 3, 1 / 5, 1 / 7, 1 / 9, 1
/ 1, 5 / 3, 5 / 5, 5 / 7, 5 / 9, 5
/ 1, 9 / 3, 9 / 5, 9 / 7, 9 / 9, 9
Corollary 2
The unit digits of and cannot be 5 and 5, 9 and 1, or 9 and 9.
Proof:
The only prime number having unit digit of 5 is 5.
It is trivial that and cannot both have unit digits of 5 because they are relatively prime.
When the unit digit of is 9, then .
And
Therefore Ω is divisible by 5.
That implies M is a multiple of 5, and so the unit digit of cannot be 1 or 9.
(Q.E.D.)
Corollary 3
If the unit digits of and are respectively 3 and 5, 5 and 1, 5 and 9, or 7 and 5, then the unit digit of must be 1 for some .
Proof:
Suppose that the unit digits of and are respectively 3 and 5, 5 and 1, 5 and 9, or 7 and 5. Then is a multiple of 5 in all four cases and by
,
we obtained for some i = 1, 2,..,n.
We now first study whether .
As p13, 5 or 7 (mod 10), we shall prove that is not divisible by 5.
When ,
When ,
When ,
Conclusively, is not divisible by 5 for and therefore
for some i = 2, 3,..,n.
Now consider i = 2, 3, …, n.
For any
is never a multiple of 5.
For any
is never a multiple of 5.
For any
is never a multiple of 5.
For any
is never a multiple of 5.
There is only one possibility left i.e.
(Q.E.D.)
The Corollaries 2 and 3 are summarized as below:
Unit digits / / / / // 1, 1 / 3, 1 / 5, 1 (**) / 7, 1 / 9, 1 (not exist)
/ 1, 5 / 3, 5 (**) / 5, 5 (not exist) / 7, 5 (**) / 9, 5
/ 1, 9 / 3, 9 / 5, 9 (**) / 7, 9 / 9, 9 (not exist)
Out of 15 possibilities, we certainly rule out 3 of them i.e. (5, 5), (9, 1), and (9, 9).
For the other four cases (3, 5), (5, 1), (7, 5) and (5, 9), we obtained the necessary condition for being odd perfect is .
The remaining cases are very difficult. So we shifted our attention to the multiples of odd perfect numbers and stepped into our second stage in which we had found some interesting properties of odd perfect numbers other than their unit digits.
Second Stage
In our first stage, we focused on the unit digits of the components p1 and M2 of odd perfect numbers, hoping to get some contradictions to the existence of odd perfect numbers. We had thought for the remaining cases for a long time without any progress, then we began to consider the multiples of odd perfect numbers. We had found some interesting results
Lemma 4:
For any positive integers a and c,
If for some positive integer b greater than 1, then .
Proof:
Let
And so
Now is multiplied by a number,
where are positive integers relatively prime to .
Let , then
Then
Theorem 2:
A multiple of a perfect number cannot be a perfect number.
Proof:
Let be a perfect number and for some natural number .
By Lemma 4, we get
is not a perfect number.
(Q.E.D.)
After proving that multiples of a perfect numbers are not perfect, we were just wondering what they are. One day, we felt and believed that they should have some properties that others don’t have. We began to explore such properties in the Third Stage.
Third Stage
In the Third Stage, we explored the relationships between perfect numbers and their multiples. Finally we succeeded to relate odd perfect numbers and 3- perfect numbers and generate higher multiply perfect numbers by lower ones. As in [1], multiply perfect numbers are defined as below :
For a given natural number k, a number n is called k-perfect (or k-multiply perfect) if and only if the sum of all positive divisors σ(n)of n is equal to kn, i.e σ(n)=kn.
Lemma 5
If is an odd perfect number, then 2 is a 3-perfect number.
Proof:
Since (2, ) =1, therefore
So 2 is a 3-perfect number.
(Q.E.D.)
Theorem 3
If the number of 3-perfect numbers is finite, then the number of odd perfect numbers is also finite.
Proof:
By Lemma 5, if there are infinitely many odd perfect numbers, then there are also infinitely many 3-perfect numbers can be generated. The theorem is proved by contrapositive.
(Q.E.D.)
Similarly, we could generate higher perfect numbers by lower ones which are relatively prime to each other.
Theorem 4
If there are n perfect numbers that are relatively prime to each other, then there exists a -perfect number.
Proof:
Let be n perfect numbers that are relatively prime to each other
and M= .
Then =
Thus M is a -perfect number.
(Q.E.D.)
Summary and Conclusions
After a long-term study, we were all satisfied with our fruitful outcomes, even though it was not perfect. However, they were all come from our sweat and effort. Our main goal is to find the general form of a hypothetical odd perfect number and eliminating those which cannot be odd perfect numbers. The following are the results of our investigation.
In the First Stage, we were glad to find that the general form of a hypothetical odd number is which is very beautiful. Using this form, modulo and divisibility, we have been more familiar with their unit digits and proved that the unit digits of and cannot be 5 and 5, 9 and 1 or 9 and 9. Furthermore, if the unit digit of is not 1 for all then the unit digits of and are never 3 and 5, 5 and 1, 5 and 9, and 7 and 5.
After investigation their general form, we started to consider the multiples of odd perfect numbers and found a beautiful inequality on the function i.e., by which we proved that any multiple of a perfect number cannot be perfect.
In the Third Stage, we continued our study by making use of the multiply perfect numbers. We found that if is an odd perfect number, then 2 is a 3-perfect number. By this result, the number of odd perfect numbers is bounded by that of 3-perfect numbers. We were all excited about this result since we may prove the non-existence of odd perfect number by proving the number of 3-perfect number is finite. Additionally, if there are n perfect numbers that are relatively prime to each other, there must a -perfect number can be generated.
After all, we hope that our investigation can have a little contribution to the advancement of Mathematics. In the coming future, we will still continue to explore the world of Mathematics!
Reference
Websites
[1] http://en.wikipedia.org/wiki/Multiply_perfect_number
[2] http://mathworld.wolfram.com/PerfectNumber.html
[3] http://mathworld.wolfram.com/OddPerfectNumber.html
[4] http://mathworld.wolfram.com/MultiperfectNumber.html
[5] http://mathworld.wolfram.com/AlmostPerfectNumber.html
[6] http://www-history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html
[7] http://www.math.niu.edu/~rusin/known-math/99/multperf
Documents
1. Dr. Ka-Leun Cheung, Number Theory and Cryptography, 2005
2. D.R. Heath-Brown, Odd Perfect numbers, Magdalen College, Oxford
3. SIMON DAVIS, A PROOF OF THE ODD PERFECT NUMBER CONJECTURE,
4. PACE P. NIELSEN, AN UPPER BOUND FOR ODD PERFECT NUMBERS,
5. JOHN VOIGHT, PERFECT NUMBERS: AN ELEMENTARY INTRODUCTION,
6. PETER HAGIS, JR. AND GRAEME L. COHEN, EVERY ODD PERFECT NUMBER HAS A PRIME FACTOR WHICH EXCEEDS 10^6,
7. DOUGLAS E. IANNUCCI, THE SECOND LARGEST PRIME DIVISOR OF AN ODD PERFECT NUMBER EXCEEDS TEN THOUSAND, 1999
8. G. G. DANDAPAT, J. L. HUNSUCKER AND ARL POMERANCE, SOME NEW RESULTS ON ODD PERFECT NUMBERS, 1975
Exploration on odd perfect numbers p.1