Primes and Greatest Common
Divisors
Dr. Delaram Kahrobaei
City Tech (CUNY)
Primes
Definition: A positive integer p greater than 1 is
called prime is the only positive factors of p are 1
and p.
Definition: A positive integer that is greater than 1
and is not prime is called composite.
Primes less than 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
The fundamental Theorem of
Arithmetic
Theorem: Every positive integer greater than 1 can
be written uniquely as a prime or as the product of
two or more primes where the prime factors are
written in order of nondecreasing size.
Examples:
100= 22 . 52
641=641
999= 33 . 37
1024= 210
A procedure for showing that an
integer is prime:
Theorem: If n is a composite integer, then n has
a prime divisor less than or equal to
.
Example: Show 101 is prime.
Solution: The only primes not exceeding
are 2, 3, 5, 7. Because 101 is not divisible by 2, 3, 5,
or 7 (the only quotient of 101 and each of these
integers is not an integer), it follows that 101 is
prime.
Theorem:
There are infinitely many primes!
There is an ongoing quest to discover larger and
larger prime numbers; for almost all the last 300
years, the largest prime known has been an
integer of the special form 2p – 1, where p is
also prime. Such primes are called Mersenne
primes.
Example: 22 -1 = 3, 23 -1 = 7, 25 -1 =31 are
Mersenne primes, while 211 -1 = 2047 is not a
Mersenne prime because 2047= 23.89.
Conjectures and Open Problems About
Primes
Large prime numbers have many applications in
cryptology.
It would be useful to have a function f(n) such
that f(n) is prime for all positive integers n.
After a lot of computation we may encounter
the polynomial f(n) = n2 –n + 41. This polynomial
has interesting property that f(n) is prime for n
integer number between 0 < n < 41.
Conjecture: f(n) = n2 –n + 41 is prime
for all integer? NO
f(1)=41
f(2)= 43
f(3)= 47
f(4)=53
BUT
f(41)= 412 -41 +41 = 412
NOT a PRIME!
Goldbach’s Conjecture
In 1742 Goldbach in a letter to Euler conjectured
that every odd integer n, n>5, is the sum of
three primes. Euler replied the conjecture is
equivalent to the conjecture that every even
integer n, n>2, is the sum of two primes.
4=2+2, 6=3+3, 8 = 5+3, 10 = 7+3, 12 = 7+5
As of 2006 the conjecture has been checked for
all positive even integer up to 2. 1017
Greatest common Divisors and Least
Common Multiple
Definition: Let a and b be integers, not both zero.
The largest integer d such that d|a and d|b is called
the greatest common divisor of a and b. The
greatest common divisor of a and b is denoted by
gcd(a,b).
Example: What is the greatest common divisor of
24 and 36?
Solution: The positive common divisors of 24 and
36 are 1, 2, 3, 4, 6, 12. Hence gcd(24,36)=12.
Relatively Prime
Definition: The integers a and b are relatively
prime if their greatest common divisor is 1.
i.e. gcd(a,b)=1.
Example: gcd (17, 22) =1 therefore 17 and 22 are
relatively prime.
The Least Common multiple
The least common multiple of the positive
integers a and b is the smallest positive integer
that is divisible by both a and b.
The least common multiple of a and b is
denoted by lcm(a, b).
Example:
lcm(233572, 2433)=2max(3,4) 3max(3,5) 7max(2,0) = 24 35 72