Perfect numbers essay

The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined when numbers of the form 2 n p 2^{n}p 2 n p, where p p p is prime, can be perfect.

Whether the five assertions of Nicomachus were based on any more than this algorithm and the fact the there were four perfect numbers known to him 6, 28, 496 and 8128, it is impossible to say, but it does seem unlikely that anything more lies behind the unproved assertions.

1 (1) The n n n th perfect number has n n n digits. 2 (2) All perfect numbers are even. 3 (3) All perfect numbers end in 6 and 8 alternately. 4 (4) Euclid 's algorithm to generate perfect numbers will give all perfect numbers i.e. ... 5 (5) There are infinitely many perfect numbers.

The number N is said to be perfect if σ(N) = 2N. When σ(N) 2N, we say N is abundant. The definition of perfect is equivalent to saying that the sum of the proper (or aliquot) divisors of N is equal to N (we just do not add N itself to the sum).