In mathematics , prime numbers with a difference of 4 are called prime cousins . For example, the numbers 13 and 17 are prime cousins because one number is 4 smaller than the other (or the other is 4 larger than one).
Prime cousins have the form . The prime cousins under 1000 are
(
p
,
p
+
4th
)
{\ displaystyle (p, p + 4)}
(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97 , 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281 ), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487 , 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827 ), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971).
(Follow A023200 in OEIS ) and (Follow A046132 in OEIS )
properties
The only prime that belongs to two pairs of prime cousins is 7. One of the numbers or is always divisible by 3, so this is the only case where the triple consists of three prime numbers.
p
,
p
+
4th
{\ displaystyle p, p + 4}
p
+
8th
{\ displaystyle p + 8}
p
=
3
{\ displaystyle p = 3}
(
p
,
p
+
4th
,
p
+
8th
)
{\ displaystyle (p, p + 4, p + 8)}
In May 2009, Ken Davis discovered the two currently largest prime cousins with 11594 digits. Of the prime cousins, the first is prime
(
p
,
p
+
4th
)
{\ displaystyle (p, p + 4)}
p
{\ displaystyle p}
p
=
(
311778476
⋅
587502
⋅
9001
#
⋅
(
587502
⋅
9001
#
+
1
)
+
210
)
⋅
587502
⋅
9001
#
-
1
35
+
1
{\ displaystyle p = (311778476 \ cdot 587502 \ cdot 9001 \ # \ cdot (587502 \ cdot 9001 \ # + 1) +210) \ cdot {\ frac {587502 \ cdot 9001 \ # - 1} {35}} + 1}
It is a Primfakultät , d. H. the product of all prime numbers .
9001
#
=
2
⋅
3
⋅
5
⋅
...
⋅
9001
{\ displaystyle 9001 \ # = 2 \ cdot 3 \ cdot 5 \ cdot \ dotsc \ cdot 9001}
≤
9001
{\ displaystyle \ leq 9001}
The greatest known prime cousins could be the following two:
(
p
,
p
+
4th
)
{\ displaystyle (p, p + 4)}
p
: =
474435381
⋅
2
98394
-
5
{\ displaystyle p: = 474435381 \ cdot 2 ^ {98394} -5}
p
+
4th
=
474435381
⋅
2
98394
-
1
{\ displaystyle p + 4 = 474435381 \ cdot 2 ^ {98394} -1}
They each have 29,629 positions and were discovered in November 2012 by Michael Angel, Paul Jobling and Dirk Augustin. The second of these two numbers,, has now been verified as a prime number, but there is currently no known prime number test that could simply determine whether the first number ,, is prime. is a PRP number ( probable prime ), so it is very likely a prime number because it fulfills conditions which all prime numbers have but which most composite numbers do not fulfill.
p
+
4th
{\ displaystyle p + 4}
p
{\ displaystyle p}
p
{\ displaystyle p}
It follows from the first Hardy-Littlewood conjecture that prime cousins have the same asymptotic density as prime twins . An analogy to Brun's constant for prime twins can also be defined for prime cousins. It is called Brun's constant for prime cousins and is the result of the convergent sum
B.
4th
=
(
1
7th
+
1
11
)
+
(
1
13
+
1
17th
)
+
(
1
19th
+
1
23
)
+
⋯
.
{\ displaystyle B_ {4} = \ left ({\ frac {1} {7}} + {\ frac {1} {11}} \ right) + \ left ({\ frac {1} {13}} + {\ frac {1} {17}} \ right) + \ left ({\ frac {1} {19}} + {\ frac {1} {23}} \ right) + \ cdots.}
The first prime number cousin pair (3, 7) is omitted.
If you insert all prime number cousins to , Marek Wolf showed in 1996 that:
2
42
{\ displaystyle 2 ^ {42}}
B.
4th
≈
1.197
0449
{\ displaystyle B_ {4} \ approx 1 {,} 1970449}
(Follow A194098 in OEIS )
This constant must not be confused with Brun's constant for prime quadruplets, which is also referred to as, but gives a different value.
B.
4th
{\ displaystyle B_ {4}}
literature
David Wells: Prime Numbers: The Most Mysterious Figures in Math . John Wiley & Sons, 2011, ISBN 1-118-04571-8 , pp. 33 .
Benjamin Fine, Gerhard Rosenberger: Number theory: an introduction via the distribution of primes . Birkhäuser, 2007, ISBN 0-8176-4472-5 , p. 206 .
Individual evidence
^ Wolfram MathWorld, cousin Primes. Retrieved December 1, 2015 .
↑ 11594 digit cousin prime pair. Retrieved December 1, 2015 .
↑ Prime pages, . Retrieved December 1, 2015 .
474435381
⋅
2
98394
-
1
{\ displaystyle 474435381 \ cdot 2 ^ {98394} -1}
↑ B.Segal: generalization you théorème de Brun . Ed .: CR Acad. Sc. URSS. Christine Steyrer, 1930, ISBN 978-3-902662-18-7 , p. 501-507 (Russian).
↑ Zentralblatt MATH Zentralblatt MATH 57.1363.06. Retrieved December 1, 2015 .
^ Marek Wolf, On the Twin and Cousin Primes ( PostScript file).
formula based
Carol ((2 n - 1) 2 - 2) |
Cullen ( n ⋅2 n + 1) |
Double Mersenne (2 2 p - 1 - 1) |
Euclid ( p n # + 1) |
Factorial ( n! ± 1) |
Fermat (2 2 n + 1) |
Cubic ( x 3 - y 3 ) / ( x - y ) |
Kynea ((2 n + 1) 2 - 2) |
Leyland ( x y + y x ) |
Mersenne (2 p - 1) |
Mills ( A 3 n ) |
Pierpont (2 u ⋅3 v + 1) |
Primorial ( p n # ± 1) |
Proth ( k ⋅2 n + 1) |
Pythagorean (4 n + 1) |
Quartic ( x 4 + y 4 ) |
Thabit (3⋅2 n - 1) |
Wagstaff ((2 p + 1) / 3) |
Williams (( b-1 ) ⋅ b n - 1)
Woodall ( n ⋅2 n - 1)
Prime number follow
Bell |
Fibonacci |
Lucas |
Motzkin |
Pell |
Perrin
property-based
Elitist |
Fortunate |
Good |
Happy |
Higgs |
High quotient |
Isolated |
Pillai |
Ramanujan |
Regular |
Strong |
Star |
Wall – Sun – Sun |
Wieferich |
Wilson
base dependent
Belphegor |
Champernowne |
Dihedral |
Unique |
Happy |
Keith |
Long |
Minimal |
Mirp |
Permutable |
Primeval |
Palindrome |
Repunit ((10 n - 1) / 9) |
Weak |
Smarandache – Wellin |
Strictly non-palindromic |
Strobogrammatic |
Tetradic |
Trunkable |
circular
based on tuples
Balanced ( p - n , p , p + n) |
Chen |
Cousin ( p , p + 4) |
Cunningham ( p , 2 p ± 1, ...) |
Triplet ( p , p + 2 or p + 4, p + 6) |
Constellation |
Sexy ( p , p + 6) |
Safe ( p , ( p - 1) / 2) |
Sophie Germain ( p , 2 p + 1) |
Quadruplets ( p , p + 2, p + 6, p + 8) |
Twin ( p , p + 2) |
Twin bi-chain ( n ± 1, 2 n ± 1, ...)
according to size
Titanic (1,000+ digits) |
Gigantic (10,000+ digits) |
Mega (1,000,000+ digits) |
Beva (1,000,000,000+ positions)
Composed
Carmichael |
Euler's pseudo |
Almost |
Fermatsche pseudo |
Pseudo |
Semi |
Strong pseudo |
Super Euler's pseudo
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