Twin primes: Difference between revisions

Latest revision as of 00:51, 23 June 2012

Definition

The term twin primes is used for a pair of odd prime numbers that differ by two. In other words, primes $p,p+2$ are termed twin primes. Either member of a pair of twin primes may be referred to as a twin prime.

The twin prime conjecture states that there are infinitely many twin primes.

Basic facts

If $p$ and $p+2$ are both prime, they both must be odd numbers.
For $p>3$ , if $(p,p+2)$ are twin primes, then $p\equiv 2{\pmod {3}}$ (hence $p\equiv 5{\pmod {6}}$ ) and $p+2\equiv 1{\pmod {3}}$ (hence $p+2\equiv 1{\pmod {6}}$ ).
In particular, with the exception of the prime $5$ , any member of a pair of twin primes cannot be a member of another pair of twin primes, i.e., the lesser member in one pair of twin primes cannot be the greater member in another pair.

Occurrence

Lesser of twin primes

The lesser of twin primes is the sequence (in increasing order) of all the primes $p$ for which $p+2$ is prime. This list goes:

3, 5, 11, 17, 29, 41, 59, 71, 101, [SHOW MORE]

View list on OEIS

Greater of twin primes

The greater of twin primes is the sequence (in increasing order) of all the primes $p$ for which $p-2$ is prime. This list goes:

5, 7, 13, 19, 31, 43, 61, 73, 103, [SHOW MORE]

View list on OEIS

Proportion of primes

$n$	Number of primes $\leq n$	Number of twin prime pairs $\leq n$	Proportion of primes that are members of twin prime pairs
10	4	2	$3/4=0.75$
100	25	8	$15/25=0.60$
1000	168	35	$69/168\approx 0.410714$
10000	1229	205	$409/1229\approx 0.332791$

Relation with other properties

Related properties for pairs of primes

Property	Meaning	Comment
Cousin primes	two primes that differ by $4$	Note that for $p>3$ , if both $p$ and $p+4$ are prime, $p+2$ is not prime. Hence, the prime gap in this case is $4$ .
Sexy primes	two primes that differ by $6$ (with no prime in between)	Since this is a pair of successive primes, the prime gap is $6$ .
Sophie Germain prime	a prime $p$ such that $2p+1$ is also prime	the corresponding prime $2p+1$ is a safe prime
safe prime	a prime $p$ such that $(p-1)/2$ is also prime	the corresponding prime $(p-1)/2$ is a Sophie Germain prime

Related properties for primes

Chen prime is a prime number $p$ such that $p+2$ is either a prime number or a semiprime. The name arises because of Chen's theorem on primes and semiprimes with fixed separation, which essentially asserts that for any fixed separation, there are infinitely many pairs of a prime and a semiprime having that separation.

Related properties for more than two primes

Property	Meaning	Comment
Prime quadruplet	a collection of four primes $p,p+2,p+6,p+8$	there can be no further primes in between
Prime constellation	a sequence of consecutive primes $p_{1}<p_{2}<\dots <p_{k}$ for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes	we are usually interested in prime constellation having a particular constellation pattern.
Bitwin chain		combines the idea of twin primes, Cunningham chain of the first kind, and Cunningham chain of the second kind

Related facts/conjectures

Broad concern	Name of fact/conjecture	Statement	Status
Infinitude	twin primes conjecture	there are infinitely many twin primes	open
Largeness, i.e., sum of reciprocals	Brun's theorem	the sum of reciprocals of all twin prime pairs (i.e., we add the reciprocal of each member of each twin prime pair) is finite	proved. This sum is Brun's constant.
Density	first Hardy-Littlewood conjecture	In the particular case of twin primes, the claim is that the number of twin prime pairs $\leq n$ is $\!\sim 2C_{2}n/(\ln n)^{2}$ , where $C_{2}$ is a specified constant.

@@ Line 1: / Line 1: @@
 ==Definition==
-The term '''twin primes''' is used for a pair of odd prime numbers that differ by two. In other words, primes <math>p, p + 2</math> are termed twin primes.
+The term '''twin primes''' is used for a pair of odd prime numbers that differ by two. In other words, primes <math>p, p + 2</math> are termed twin primes. Either member of a pair of twin primes may be referred to as a '''twin prime'''.
-The [[twin primes conjecture]] states that there are infinitely many twin primes.
+The [[twin prime conjecture]] states that there are infinitely many twin primes.
+==Basic facts==
+* If <math>p</math> and <math>p + 2</math> are both prime, they both ''must'' be odd numbers.
+* For <math>p > 3</math>, if <math>(p,p+2)</math> are twin primes, then <math>p \equiv 2 \pmod 3</math> (hence <math>p \equiv 5 \pmod 6</math>) and <math>p + 2 \equiv 1 \pmod 3</math> (hence <math>p + 2 \equiv 1 \pmod 6</math>).
+* In particular, with the exception of the prime <math>5</math>, any member of a pair of twin primes cannot be a member of ''another'' pair of twin primes, i.e., the lesser member in one pair of twin primes cannot be the greater member in another pair.
+==Occurrence==
+===Lesser of twin primes===
+The ''lesser of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p + 2</math> is prime. This list goes: <section begin="list-lesser"/>[[3]], [[5]], [[11]], [[17]], [[29]], [[41]], [[59]], [[71]], [[101]], <toggledisplay>107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607</toggledisplay>[[oeis:A001359|View list on OEIS]]<section end="list-lesser"/>
+===Greater of twin primes===
+The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: <section begin="list-greater"/>[[5]], [[7]], [[13]], [[19]], [[31]], [[43]], [[61]], [[73]], [[103]], <toggledisplay>109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609</toggledisplay>[[Oeis:A006512|View list on OEIS]]<section end="list-greater"/>
+===Proportion of primes===
+{| class="wikitable" border="1"
+! <math>n</math> !! Number of primes <math>\le n</math> !! Number of twin prime pairs <math>\le n</math> !! Proportion of primes that are members of twin prime pairs
+|-
+| 10 || 4 || 2 || <math>3/4 = 0.75</math>
+|-
+| 100 || 25 || 8 || <math>15/25 = 0.60</math>
+|-
+| 1000 || 168 || 35 || <math>69/168 \approx 0.410714</math>
+|-
+| 10000 || 1229 || 205 || <math>409/1229 \approx 0.332791</math>
+|}
+==Relation with other properties==
+===Related properties for pairs of primes===
+{| class="wikitable" border="1"
+! Property !! Meaning !! Comment
+|-
+| [[Cousin primes]] || two primes that differ by <math>4</math> ||  Note that for <math>p > 3</math>, if both <math>p</math> and <math>p + 4</math> are prime, <math>p+2</math> is not prime. Hence, the prime gap in this case is <math>4</math>.
+|-
+| [[Sexy primes]] || two primes that differ by <math>6</math> (with no prime in between) || Since this is a pair of successive primes, the prime gap is <math>6</math>.
+|-
+| [[Sophie Germain prime]] || a prime <math>p</math> such that <math>2p + 1</math> is also prime || the corresponding prime <math>2p + 1</math> is a safe prime
+|-
+| [[safe prime]] || a prime <math>p</math> such that <math>(p - 1)/2</math> is also prime || the corresponding prime <math>(p - 1)/2</math> is a Sophie Germain prime
+|}
+===Related properties for primes===
+* [[Chen prime]] is a [[prime number]] <math>p</math> such that <math>p + 2</math> is either a prime number or a [[semiprime]]. The name arises because of [[Chen's theorem on primes and semiprimes with fixed separation]], which essentially asserts that for any fixed separation, there are infinitely many pairs of a prime and a semiprime having that separation.
+===Related properties for more than two primes===
+{| class="wikitable" border="1"
+! Property !! Meaning !! Comment
+|-
+| [[Prime quadruplet]] || a collection of four primes <math>p,p+2,p+6,p+8</math> || there can be no further primes in between
+|-
+| [[Prime constellation]] || a sequence of consecutive primes <math>p_1 < p_2 < \dots < p_k</math> for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes || we are usually interested in prime constellation having a particular constellation pattern.
+|-
+| [[Bitwin chain]] || || combines the idea of twin primes, [[Cunningham chain of the first kind]], and [[Cunningham chain of the second kind]]
+|}
+==Related facts/conjectures==
+{| class="wikitable" border="1"
+! Broad concern !! Name of fact/conjecture !! Statement !! Status
+|-
+| Infinitude || [[twin primes conjecture]] || there are infinitely many twin primes || open
+|-
+| Largeness, i.e., sum of reciprocals || [[Brun's theorem]] || the sum of reciprocals of all twin prime pairs (i.e., we add the reciprocal of each member of each twin prime pair) is finite || proved. This sum is [[Brun's constant]].
+|-
+| Density || [[first Hardy-Littlewood conjecture]] || In the particular case of twin primes, the claim is that the number of twin prime pairs <math>\le n</math> is <math>\! \sim 2C_2n/(\ln n)^2</math>, where <math>C_2</math> is a specified constant.
+|}