A Bit of UML
A Mersenne prime, then, is a special type of prime number. In UML, this is represented as an is a relationship: a Mersenne prime is a special type of prime number. And since k in 2k-1 is also prime, then there is also a has a relationship: a Mersenne number has a prime number.
From the Euler theorem on perfect numbers above, it is clear that there are two types of perfect numbers -- odd and even. Again, this can be represented in UML as follows:
It is interesting to note that an odd perfect number has never been found, even though some extraordinary properties about them are known, for instance:
- An odd perfect number cannot be divided by 105.
- An odd perfect number must contain at least eight different prime factors.
- The smallest odd perfect number must exceed 10300.
- The second largest prime factor of an odd number exceeds 1,000.
- The sum of the reciprocals of all odd perfect numbers is finite.
Check out item three; imagine trying to do some long division sums in your head with that number.
Finally, again from the Euler theorem, perfect numbers are related to prime numbers in that an even perfect number can be expressed in the form n = 2k-1(2k-1); i.e., n = 2k-1 (a Mersenne number). Therefore, every even perfect number has a Mersenne number. Once more, in UML this can be diagrammatically depicted as follows: