Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when "n" = 11, the fifth prime. However, 2 - 1 = 2047 = 23 × 89 is not prime and therefore "n" = 11 does not give a perfect number. Two other wrong assumptions were:
The fifth perfect number (formula_1) has 8 digits. This falsifies the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It is straightforward to show the last digit of any even perfect number must be 6 or 8.
In order for formula_2 to be prime, it is necessary that formula_3 should be prime. Prime numbers of the form 2 - 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.
Two millennia after Euclid, Euler proved that the formula 2(2 - 1) will yield all the even perfect numbers. Therefore, every Mersenne prime will yield a distinct even perfect number–there is a concrete one-to-one association between even perfect numbers and Mersenne primes. This result is often referred to as the "Euclid-Euler Theorem". Till January 2013, only 48 Mersenne primes are known. This means there are 48 perfect numbers known, the largest being 2 × (2 - 1) with 34,850,340 digits.
The first 42 even perfect numbers are 2(2 - 1) for
The other 7 known are for "n" = 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281.
It is currently not known whether there are others between them.
It is still not known if there are infinitely many Mersenne primes and perfect numbers. The search for new Mersenne primes is the goal of the GIMPS distributed computing project.
Since any even perfect number has the form 2(2 - 1), it is a triangular number, and, like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2 - 1. Also, any even perfect number except the first one is the sum of the first 2 odd cubes:
It is not known whether there are any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. Also, it has been conjectured that there are no odd Ore's harmonic numbers. If true, this would mean that there are no odd perfect numbers.
Any odd perfect number "N" must satisfy the following conditions:
Let formula_10 be odd perfect number. Since divisor function is multiplicative,
formula_11.
formula_12 must be an even not divisible by 4 and all the remaining must be odd.
formula_13
forces formula_14.
Even perfect numbers have a very precise form; odd perfect numbers are rare, if indeed they do exist. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's Strong Law of Small Numbers:
The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with "perfect" itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.
By definition, a perfect number is a fixed point of the restricted sum-of-divisors function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.
StarCraft
StarCraft is a real-time strategy game for the PC. It was created in 1998 by Blizzard Entertainment. "Starcraft" is 14 years old and is still played by many people, in many countries around the world. It is very popular in countries like Korea and the United States.
"StarCraft" and its expansion pack "StarCraft: Brood War" sold more than 1.5 million copies in 1998, making it the best selling game of the year. In 1998, "StarCraft" won an Origins Award for best computer game.
On May 19, 2007 Blizzard announced that they are making StarCraft II. Also, they have held some events for South Korean "StarCraft" pro-gamers to help them create the game.
The story is set in the future, where humans (called "Terrans") get stuck in a war between two alien species, the "Zerg" and the "Protoss". The Zerg look like monsters, and are all controlled by a very powerful Zerg called the "Overmind". They fly around the universe and invade planets. The Protoss, who have very advanced technology, look for the planets that the Zerg have taken over. When they find a Zerg planet, they burn it so that the Zerg on that planet cannot spread to others. In the original series human protagonist Jim Raynor and several Protoss allies attack and defeat the Overmind. In "Starcraft: Brood War" Jim's ex-partner Sarah Kerrigan uses deceit and trickery to manipulate the Terran and Protoss into helping her take control of the entire Zerg brood, making her the strongest force in the galaxy.
"StarCraft" players can choose to play as Protoss, Zerg, or Terran races. Each race has different soldiers and buildings. Players can choose to play one-player or multiplayer games. In both, the player must gather resources, and use them to build a base and make soldiers to attack the enemy. The player wins when they destroy all the enemies buildings.
In one-player mode, there is the Campaign mode, and Custom mode. In Campaign mode, the player goes through a series of missions, which tell the story. Custom mode lets a player choose what race they want to be, what area they want to play on, and what race they will fight against (all enemies are controlled by the computer).
During a game, the player must use workers to gather minerals and vespene gas. Minerals appear as blue crystals in the ground and vespene gas comes from a geyser that has green smoke coming out of it. To gather vespene gas, the player needs to build a refinery building to be built on the geyser. The player uses minerals and vespene gas they have gathered to construct buildings and create units to fight the enemy. Buildings are structures that are used to create different kinds of units to fight the enemy. In order to create units, players must build special structures or units that provide "supply". If the player does not have enough supply, they cannot create more units. Each player has a supply limit of 200 units. When the player has reached 200 units, they can not make more units.
There are many different buildings the player can construct. Most of the buildings are used to build units. There are other buildings that are used for research to upgrade units, making them more powerful. Players can also make defensive buildings that are used to defend against attacking enemy units.
Each race have different types of units. The Protoss have powerful warriors and machinery that cost lots of minerals and gas to build. The Zerg use many fast and low cost units to defeat their enemies. Terran units mix between Protoss and Zerg units. They have strong, but easy to kill units that are powerful when used in formation. No race in "StarCraft" is more powerful than another race. Blizzard has released many small updates to the game that keep all the races balanced.