By: Christina Mantas

Edited and Extended by:

Nicole Gillum

A prime number is a positive integer that

has exactly two factors. • So 1 is not prime, 2 is, and 3 is and 4 isn‟t and so

fourth

• 2 is the smallest possible difference between

two primes

The question of whether or not there are infinitely many twin primes is one of the most elusive and longest-standing open questions in number theory.

A Twin Prime is a pair of primes that differ by two.• Except for the pair (2, 3), 2 is the smallest possible

difference between two primes.

• Examples of Twin Primes are: (3,5),(5,7),(11,13),(17,19),(29,31),…,(821,823)

In 1849, Alphonse de Polignac, a French

mathematician formulated what is know

as Polignac‟s Conjecture• Polignac‟s Conjecture states, “For every natural

number k, there are infinitely many prime pairs

p and q such that p-q=2k."

• “There are infinitely many „prime gaps‟ of size

2k” The „prime gaps‟ meaning the difference

between two successive prime numbers.

The Twin Prime Conjecture is the “k=1”

case of Polignac‟s Conjecture

It states, “There are infinitely many primes

p such that p+2 is also prime”

The difference between two primes is

called the “prime gap”

The first 30 prime gaps are:

[1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,

6,8,4,2,4,2,4,14]• These represent the difference between (2, 3),

(3, 5), (5, 7), and so forth.

Pairs of primes with prime gap = 4 are

called “Cousin Primes”• Examples: (3, 7), (7, 11), (13, 17), and so on

Pairs of primes with prime gap = 6 are

called “Sexy Primes”• Examples: (5, 11), (7, 13), (11, 17) and so on

The last largest known Twin Prime was

found January 15, 2007 by Eric Vautier, a

French mathematician who submitted his

solution to the two distributed computing

projects, The Twin Prime Search and

Prime Grid. The number is:• 2003663613* 2 to the power of 195000 +/- 1

• The numbers have 58711 decimal digits

The current largest known Twin Prime

was found on August 6, 2009 by a team of

two computer programs and two

researchers. It is:• 65516468355·2333333 +/- 1

• The numbers have 100,355 decimal digits

The largest known Cousin Primes were

discovered in May of 2009 by Ken Davis• p =

(311778476·587502·9001#·(587502·9001#+1)+2

10)·(587502·9001#−1)/35+1

where 9001# is called a “primorial”

• This number has 11594 digits

“9001#” is referred to as “9001 primorial”

A primorial works like a factorial but with

the composite factors removed from the

product.

The primorial is the product of only the

prime numbers leading up to the number. Example, 9001# = 2*3*5*7*11*13*17*…*9001

Notably, Ken Davis also discovered the

largest known Sexy Prime in May, 2009• p =

(117924851×587502×9001#×(587502×9001#+1)

+210)×(587502×9001#−1)/35+5

• it has 11593 digits