Hello, pantyman-ga
The old science major that I am(Chemistry), I walked into this one
with a bit of perspiration(until my air conditioning activated). It
turned out to be a delightful jump into the mystery of that "odd
perfect number".
I do hope you enjoy these sites and the discussions.
1. Using a search term of "odd perfect numbers", a number of websites
popped up.
The first is entitled: "Perfect Numbers" and offers a historical
review of this area.
"It is not known when perfect numbers were first studied and indeed
the first studies may go back
to the earliest times when numbers first aroused curiosity"
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html
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2. The next site has a short statement and starts thus:*
Does there exist a number that is perfect and odd?
"This question was first posed by Euclid in ancient Greece. This
question is still open.."
"A given number is perfect if it is equal to the sum of all its proper
divisors..."
"This question was first posed by Euclid in ancient Greece. This
question is still open.."
http://www.cs.unb.ca/~alopez-o/math-faq/node60.html
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3.The Perfect Number Journey
http://home1.pacific.net.sg/~novelway/MEW2/lesson1.html
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4. On the nonexistence of an odd perfect number, this site has 3
discussion messages emanating from a professor
in Manila which captures the latest efforts on Euclid's delimma of the
"perfect odd number".
"...Bad news this time to fellow number theorists around the world!
What I have just apparently shown is NOT the absolute nonexistence of
odd perfect numbers,
but just certain restrictions to their existence. The following are
the restrictions..."
http://www.askdrmath.com/epigone/sci.math.research/swoxgendphol
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5. Proof that odd perfect number does NOT exist by Eddie Wu
First we have to prove that If 2^k - 1 is prime and if N = 2^(k-1)
times 2^k - 1, then N is perfect.
The proof is shown in the book I borrowed from the library.
I just state here as follows: Let p = 2^k - 1 be the prime number.
The proper divisors of N must themselves contain only the primes 2 and
p.
Then sum of porper divisors of N = 1 + 2 + 4 + ... + 2^(k-1) + p +2p +
4p + ... + (2^(k-2))p =
(1+2+4+...+2^(k-1)) + p(1+2+4+...+2^(k-2)) = (2^k-1) + p(2^(k-1)-1) =
p + p2^(k-1) -p = p2^(k-1) =
N Because Euclid's number N equals the sum of its proper divsors, it
is perfect. Q.E.D.
In order to prove the odd perfect number does not exist, we just need
to prove that N must be even.
Here is my proof. 1. odd number + odd number = even number 2. odd
number + even number =
odd number 3. even number + even number =
even number 4. any number which is multiple of 2 must be even number
N=1+2+4+...+2^(k-1) + p+2p+4p+...+(2^(k-2))p = 1+(2+4+...+2^(k-1)) + p
+ (2p+4p+...+(2^(k-2))p) =
1 + (even number) + p + (even number) = 1 + p + (even number + even
number) =
1 + p + (even number) = p + (1 + even number) = p + odd number =
odd number + odd number ( p is a prime and must be odd number) = even
number Q.E.D.
http://mam2000.mathforum.org/epigone/sci.math.num-analysis/yaxworstin/qpk4nn24g06d@forum.mathforum.com
Well...there it is. Mr Wu is certainly sure that he put Euclid's
nightmare to bed!
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6. I've saved the best site until last.
It's Wolfram's website and the title is:
"Odd Perfect Number".
Enjoy...anything connected to Wolfram has got to be exquisite as was
his latest book.
(A New Kind of Science).
"...To this day, it is not known if any odd perfect numbers exist,
although numbers up to 10300 have been checked without success..."
Please do read this last short but very clear article on the latest
for that "odd perfect number".
http://mathworld.wolfram.com/OddPerfectNumber.html
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Brad-ga |
