Sociable and Amicable Numbers ....

Number theory is one of the most intriguing branches of mathematics. Three weeks ago, I ran into Ken Ribet, who was a contributor to Andrew Wiles' proof of Fermat's last theorem . I've sat in a few of Ken's classes, and his clarity of thought astounds me. Anyway, that kinda inspired me to think about some interesting 'early' number theory 'results'

1. Prime numbers: Nothing fancy about these little fellerz, except that for centuries, noone was able to determine an algorithmic way to obtain prime numbers. Even now, with the development of computers, although there are many algorithms to check primeness, there isnt any analytical method for obtaining prime numbers ad infinitum .

2. Perfect numbers: A perfect number is a cycle of length 1 of s, i.e., a number whose positive divisors (except for itself) sum to itself. The smallest such number is 6 : the divisors of six are 1,2,3, which add up to 6. The second number is 28 (1,2,4,7,14) whose divisors add up to 28. The third such number is 496. Interestingly, there is a hypothesis that claims that all perfect numbers are EVEN, and that there are an infinite number of them. A conjecture yet to be proved/disproved!

3. Amicable/Sociable numbers : An amicable "pair" of numbers is a cycle of length 2 of s., i.e., a pair of numbers each of which equals the sum of the other positive divisors; the members of amicable pairs are also called amicable. The smallest such pair is (220,284).

(sum of all the divisors of 220) 1+2+4+5+10+11+20+22+44+55+110 = 284
(sum of all the divisors of 284) (too tedious and boring to write down) = 220

Sociable numbers are sets if numbers with cycle > 2.

4. Fibonacci Numbers : This is perhaps the most commonly known of all number sequences. Fibonacci numbers are the numbers in the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . , each of which, after the second is the sum of the two previous ones. They are found in a variety of problems, mathematical and nonmathematical. Leaves are arranged on a stem in fibonacci sequences, and often flowers have numbers of petals equal to fibonacci numbers. Much has been said and explored about these sequences, which is why we mathematicians dont find it interesting anymore :D

5. The number of the beast "666" , as any Iron Maiden fan will know, is known as the number of the beast. It has some very very strange properties indeed! To list a few:

666 = 6 + 6 + 6 + 6³ + 6³ + 6³
666 = (6 + 6 + 6) · (6² + 1²)
666 = 6! · (6² + 1²) / (6² + 2²)
The sum of the squares of the first 7 primes is 666:

666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²
The sum of the first 144 (= (6+6)·(6+6)) digits of pi is 666.

There is a lot more .. if you are really nice to me, maybe I'll tell you some more ;)