Difference between revisions of "Perfect number"

Revision as of 11:41, 29 August 2006

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

The first four perfect numbers are 6, 28, 496, and 8128. These were the only perfect numbers known to ancient mathematicians.

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

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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-A number $n$ is a '''perfect number''' if it is the sum of its '''proper''' divisors.
+A number <math>n</math> is a '''perfect number''' if it is the sum of its '''proper''' divisors.
 The first four perfect numbers are 6, 28, 496, and 8128. These were the only perfect numbers known to ancient mathematicians.
-'''Theorem''': $n$ is an even perfect number iff $n=\frac{p(p+1)}{2}, where $p$ is a prime number equal to $2^k-1$ for some $k$.
+'''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>.
-Primes of the form $2^k-1$ are called [[Mersenne prime]]s.
+Primes of the form <math>2^k-1</math> are called [[Mersenne prime]]s.
 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.

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Difference between revisions of "Perfect number"

Revision as of 11:41, 29 August 2006