Perfect Numbers

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  • Topic: Perfect number, Prime number, Mersenne prime
  • Pages : 3 (843 words )
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  • Published : November 10, 2010
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Perfect numbers
Mathematicians have been fascinated for millenniums by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. A perfect number is a whole number, an integer greater than zero and is the sum of its proper positive devisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, including itself. For example; 6, the first perfect number, its factors are 1, 2, 3 and 6. So, 1 + 2 + 3 = 6 or alternatively (1+2+3+6) divided by 2, which = 6. The next perfect number is 28, and its factors are 1, 2, 4, 7, 14, and 28. So, if we use the method which excludes the perfect number 28, we express the equation as, 1+2+4+7+14 = 28. Or using the other method, (1+2+4+7+14+28) divided by 2, it results as 28, thus making 28 a perfect number.

Throughout history, there have been studies on perfect numbers. It is not clarified when perfect numbers were first studied and it’s possible that the first studies may go back to the earliest era when numbers first aroused curiosity. It is assumed, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked. Perfect numbers were studied by Pythagoras and his followers, mainly for their mystical properties than for their number theoretic properties. Although, the four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries. The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid's Elements written around 300BC. Also, it’s quite interesting to know that, not all perfect numbers were found in the same order, the ninth, tenth, and eleventh perfect numbers were found after the twelfth was discovered. The 29th was found...
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