Number Theory
Definitions:
factor (divisor) – If a and b are any integers with a not equal to 0, then a
is a divisor of b if and only if there is an integer x such that ax = b
proper factors – the factors of a number that are less than the number
itself
multiple – If a and b are any integers with a not equal to 0, then b is a
multiple of a if and only if there is an integer x such that ax = b
greatest common factor (GCF) – the GCF of two integers a and b is the
greatest integer that is a factor of both a and b
least common multiple (LCM) – the LCM of two positive integers is the
least positive multiple that the two numbers have in common
prime number – a counting number with exactly two different factors,
itself and 1
composite number – a number that has more than two different factors
(The number 1 is neither prime not composite.)
prime factorization – the expression of a number as the product of the
powers of its prime factors
perfect – a number is perfect if the sum of its factors (other than the
number itself) is equal to the number
abundant – a number is abundant if the sum of its factors is greater than
the number itself
deficient – a number is deficient if the sum of its factors is less than the
number itself
relatively prime (also called strangers) – two counting numbers are
relatively prime if the greatest common factor of the two numbers is 1
twin primes – two consecutive primes whose difference is 2
triple primes – three consecutive primes with differences of 2 between
the consecutive primes
cousin primes – a pair of primes of the form (p, p+4), examples are: (3,7)
and (7,11)
sexy primes - a pair of primes of the form (p, p+6), (``sexy'' since
``sex'' is the Latin word for ``six.'') The first few sexy prime pairs are
(5, 11), (7, 13), (11, 17), (13, 19), (17, 23).
Germain prime (named for Sophie Germain) – a prime p such that 2p + 1 is
also a prime
palindrome – a number that is unchanged when its digits are reversed
amicable – a pair of natural numbers m and n are amicable if the sum of
the proper divisors of m equals n and the sum of the proper divisors of n
equals m
betrothed – two numbers are said to be betrothed if the sum of all
proper factors greater than 1 of one number equals the other and vice
versa
Smith number (discovered by Harold Smith) – a counting number the sum
of whose digits is equal to the sum of all the digits of its prime factors