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Perfect Numbers

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by

Harish Kang

on 16 December 2013

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Transcript of Perfect Numbers

An introduction to Perfect Numbers...
Our presentation is on perfect numbers.
When were Perfect Numbers discovered?
Perfect numbers were first discovered by Ancient Greek mathematicians. One of them, Nicomachus, found the up to the fourth perfect number.
Later on, up to the seventh perfect number was found.
Euclid's Work in Perfect Numbers
Euclid, a famous mathematician, discovered a formula to work out a perfect number.
The formula was (2^(P-1))(2^p−1) and the solution is always an even perfect number. It works if P is a prime number.
By Harish Kang, Joel J & Derek Zhao
Odd Perfect Numbers
At the moment, no odd perfect numbers have been found, and some mathematicians have concluded that there are none.
However people are working to try and see if there are any, as people have figured out certain complex properties that the number must have.
Perfect Numbers...
The Formula In Action
A perfect number is one of the following two things, which are both equivalent to each other:
"Equal to the sum of its positive divisors(when excluding itself)" and:"Equal to half the sum of its positive divisors (including itself)"
The fifth perfect number was discovered by an unknown mathematician, and the sixth and seventh had been discovered by an Italian mathematician.
1st 6
2nd 28
3rd 496
4th 8128
5th 33550336
6th 8,589,869,056
7th 137,438,691,328
More even perfect numbers.
The Fifth, Sixth, and Seventh
The perfect numbers
for p = 2: 2^1(2^2−1) = 6
for p = 3: 2^2(2^3−1) = 28
for p = 5: 2^4(2^5−1) = 496
for p = 7: 2^6(2^7−1) = 8128.
Here are some examples
of the formula being
used.
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. It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked. Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
More History
Full transcript