What is Perfect number?
A [[number]] is called a '''perfect number''' if by adding all the positive [[divisor]]s of the number (except itself), the result is the number itself. Six is the first perfect number. Its divisors (other than the number itself: six) are one, two, and three and one plus two plus three equals six. Other perfect numbers include 28, 496 and 8128. ==Perfect numbers that are even== [[Euclid]] discovered that the first four perfect numbers are generated by the formula 2<sup>''n''-1</sup>(2<sup>''n''</sup> - 1): :for ''n'' = 2: 2<sup>1</sup>(2<sup>2</sup> - 1) = 6 :for ''n'' = 3: 2<sup>2</sup>(2<sup>3</sup> - 1) = 28 :for ''n'' = 5: 2<sup>4</sup>(2<sup>5</sup> - 1) = 496 :for ''n'' = 7: 2<sup>6</sup>(2<sup>7</sup> - 1) = 8128 Euclid saw that 2<sup>''n''</sup> - 1 is a [[prime number]] in these four cases. He then proved that the formula 2<sup>''n''-1</sup>(2<sup>''n''</sup> - 1) gives an even perfect number whenever 2<sup>''n''</sup> - 1 is prime (Euclid, Prop. IX.36). 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 two, three, five, and seven are precisely the first four primes, the fifth perfect number would be obtained when ''n'' = 11, the fifth prime. However, 2<sup>11</sup> - 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 would have five digits since the first four had one, two, three, and four digits respectively. *The perfect numbers would alternately end in six or eight. The fifth perfect number (<math>33550336=2^{12}(2^{13}-1)</math>) has eight digits. This falsifies the first assumption. For the second assumption, the fifth perfect number indeed ends with a six. However, the sixth (8,589,869,056) also ends in a six. It is straightforward to show the last digi