The
Fundamental
Theorem of Arithmeticstates that every
positive integer (except the number 1) can be represented in exactly
one way
as a product of
one or more primes.
The theorem establishes the importance
of prime numbers. The prime
numbers are the basic building blocks of the positive integers, in the
sense that every positive integer can be constructed from primes, and
there is essentially only one such construction.
The proper divisors of N are all the divisors of N excluding N
itself. (eg) 1, 2, 3 are the proper divisors of 6
An abundant number is a composite number whose divisors (excluding
the number itself) have a sum greater than the number. For example, 12 has
factors of 1, 2, 3, 4, 6, and 12. The sum 1+2+3+4+6 = 16, and 16 > 12.
A perfect number is one whose factors are equal to a given number. So,
6 is perfect because 1 + 2 + 3 = 6. Euclid (3rd c. BC) was the first to mention
perfect numbers. By the 1st c. AD, Nicomachus knew the first four (which
you are asked to find; the fifth is 33,550,336).
A deficient number is a composite number in which the sum of its factors
is less than the given number. For example, the number 8 has factors (divisors)
of 1, 2, 4, and 8. If you disregard 8 as a factor, then the sum 1 + 2 + 4
= 7 and 7 is less than 8. Therefore, 8 is deficient.
Two positive integers are said to be buddies, friends, or amicable
pairs if each one is equal to the sum of the proper divisors
of the other. For example, 220 and 284 are buddies since:
divisors of 220: 1+2+4+5+10+11+20+22+44+55+110 = 284
divisors of 284: 1+2+4+71+142 = 220
The Fibonacci
Sequence begins 1, 1, 2, 3, 5, 8, 13, 21, ..., where each term is the
sum of the two terms preceding it. Mathematicians define this sequence
recursively as follows: