Perfect Number: What Makes Him Perfect?
What kind of a man do you prefer? And what guy do you regard as a perfect man? Mathematics has a distinct view as it define what as perfect. A perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself. (from Wkidipedia) ; 6 is perfect number, because 1, 2 and 3 which are positive divisors of 6 sums up to 6. Today, people define a perfect number in terms of its divisors, but in the early times, they had seen in terms of the 'aliquot parts' of a number. An aliquot part of a number means a quotient of the number; in the problem 6 ÷ 3, the quotient would be 2 –from Wkidipedia.
It is not known exactly beginning of perfect numbers. But, some professional assume that Egyptians would have access to the numbers naturally with use of their methods of calculation. After, the numbers were studied by Pythagorians.
Interestingly, there exists only three perfect numbers between 0 and 1000: 6, 28, and 496. of course, many perfect numbers over 1000 are being finded: 8128, 33550336, 8589869056,
6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14,
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
Euclid discovered that the first four perfect numbers are generated by the formula 2n−1(2n−1), and moreover he proved that the formula 2n−1(2n−1) only gives an even perfect number if 2n−1is prime ( Euclid , Prop. IX.36). Prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself (from Wikipedia.)
for n = 2: 21(22 − 1) = 6
for n = 3: 22(23 − 1) = 28
for n = 5: 24(25 − 1) = 496
for n = 7: 26(27 − 1) = 8128
Besides 1+2+3=6 as a characteristic of perfect number, the number 6 involve many special meanings. Besides 1+2+3=6 as a characteristic of perfect number. More interestingly, the Bible said that Jesus Christ created the whole world in ‘6 days’.
According to Euclid ’s calculation, perfect number must be a even number, but perfect number don’t have to be a even number. And another question is that the number of perfect number is unlimited, but these problems have been unsolved over 2,000 years, yet.
Friendship Proven by Number, of Number
(220, 284), (17296, 18416), (1184, 1210)
These pairs have been considered as some friendly relationships. Amicable numbers are two different numbers so related that the sum of the proper divisors of the one is equal to the other, one being considered as a proper divisor but not the number itself. Such a pair is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220. These were known to the Pythagoreans, and the most famous fair 220 and 284 has regarded as a symbol of friendship. Another pair 17296 and 18416 was named by Fermat, the other pair 1184 and 1210 by Decartes. It is traditionally said that a love is forever if he himself had one fruit and gave the other fruit to his honey once he put 220 and 284 on each two fruits, by one Arab fortuneteller. Furthermore, in the Bible, the number of goats Jacob gave to his brother Esau ("two hundred she-goats and twenty he-goats") and the number of sheep ("two hundred ewes and twenty rams," Genesis 32:14) is each 220. The number 220 in Lashon Ha-Kodesh(ancient form of Ivrith) means affection or love, and the counterpart 284 means that ‘make the bead for me.’
12496, 14288, 15472, 14536, 14264
In the 20th century, mathematicians tried expanding the thought of amicable numbers. Ultimately, they found out several groups made of numbers more than 2. The members of this group is more than friendship such as 220 and 284. First, divisors of the first number 12496 sums up 14288, and divisors of 14288 sums up to 15472. Following this order, divisors of 14264 make a total sum to 12496, finally. These kinds of numbers known to us, the number of the largest group is 28, and in the group, the first number is 14316.