Difference between revisions of "Perfect number"
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| − | A | + | A [[positive integer]] <math>n</math> is called a '''perfect number''' if it is the sum of its [[proper divisor]]s. |
| − | The first four perfect numbers are 6 | + | 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> | ||
| + | * <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. | ||
| + | Some other perfect numbers include: | ||
| + | * <math>33550336</math> | ||
| + | * <math>8589869056</math> | ||
| + | * <math>137438691328</math> | ||
| + | * <math>2305843008139952128</math> | ||
| − | |||
| − | |||
| − | 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. | + | ==Even Perfect Numbers== |
| + | <math>n</math> is an [[even integer | 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. | ||
| + | |||
| + | ===Proof=== | ||
| + | |||
| + | |||
| + | |||
| + | It is conjectured that there are are infinitely many Mersenne primes and hence infinitely many even perfect numbers. No [[odd integer | 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. | ||
| + | |||
| + | |||
| + | Numbers which are not perfect may be either [[deficient number]]s or [[abundant number]]s. | ||
| + | |||
| + | ==Resources== | ||
| + | http://mathforum.org/dr.math/faq/faq.perfect.html | ||
Latest revision as of 18:09, 3 March 2022
A positive integer 
is called a perfect number if it is the sum of its proper divisors.
The first four perfect numbers are:
These were the only perfect numbers known to ancient mathematicians. Some other perfect numbers include:
Even Perfect Numbers
is an even perfect number if and only if 
, where 
is a prime number of the form 
for some 
.
Primes of the form 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 
. It is conjectured that there are none. No one has been able to prove or disprove these conjectures.
Numbers which are not perfect may be either deficient numbers or abundant numbers.
