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Difference between revisions of "Perfect number"

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A number <math>n</math> is a '''perfect number''' if it is the sum of its '''proper''' divisors.
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A [[positive integer]] <math>n</math> is called a '''perfect number''' if it is the sum of its [[proper divisor]]s.
  
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The first four perfect numbers are 6, 28, 496, and 8128. These were the only perfect numbers known to ancient mathematicians.  
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The first four perfect numbers are <math>6 = 3 + 2 + 1</math>, <math>28 = 14 + 7 + 4 + 2 + 1</math>, <math>496 = 248 + 124 + 62 + 31 + 16 + 8 + 4 + 2 + 1</math>, and <math>8128 = 4064 + 2032 + 1016 + 508 + 254 + 127 + 64 + 32 + 16 + 8 + 4 + 2 + 1</math>. These were the only perfect numbers known to ancient mathematicians.
  
  
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'''Theorem''': <math>n</math> is an even perfect number iff <math>n=\frac{p(p+1)}{2}</math>, where <math>p</math> is a prime number equal to <math>2^k-1</math> for some <math>k</math>.
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==Even Perfect Numbers==
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<math>n</math> is an even perfect number if and only if <math>n=\frac{p(p+1)}{2}</math>, where <math>p</math> is a [[prime number]] of the form <math>2^k-1</math> for some <math>k</math>.
 
Primes of the form <math>2^k-1</math> are called [[Mersenne prime]]s.  
 
Primes of the form <math>2^k-1</math> are called [[Mersenne prime]]s.  
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===Proof===
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It is conjectured that there are are infinitely many Mersenne primes and hence infinitely many even perfect numbers. No odd perfect numbers are known, and any that do exist must be greater than <math>10^{500}</math>. It is conjectured that there are none. No one has been able to prove or disprove these conjectures.
 
It is conjectured that there are are infinitely many Mersenne primes and hence infinitely many even perfect numbers. No odd perfect numbers are known, and any that do exist must be greater than <math>10^{500}</math>. It is conjectured that there are none. No one has been able to prove or disprove these conjectures.

Revision as of 10:35, 7 September 2006

A positive integer $n$ is called a perfect number if it is the sum of its proper divisors.

The first four perfect numbers are $6 = 3 + 2 + 1$, $28 = 14 + 7 + 4 + 2 + 1$, $496 = 248 + 124 + 62 + 31 + 16 + 8 + 4 + 2 + 1$, and $8128 = 4064 + 2032 + 1016 + 508 + 254 + 127 + 64 + 32 + 16 + 8 + 4 + 2 + 1$. These were the only perfect numbers known to ancient mathematicians.


Even Perfect Numbers

$n$ is an even perfect number if and only if $n=\frac{p(p+1)}{2}$, where $p$ is a prime number of the form $2^k-1$ for some $k$. Primes of the form $2^k-1$ are called Mersenne primes.

Proof

It is conjectured that there are are infinitely many Mersenne primes and hence infinitely many even perfect numbers. No odd perfect numbers are known, and any that do exist must be greater than $10^{500}$. It is conjectured that there are none. No one has been able to prove or disprove these conjectures.

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