PRIME QUADRUPLETSMathematics
Number Theory
By Megan Duke – Muskingum University
Review
Prime – a natural number great than 1 that has no positive divisors other than 1 and itself.
Quadruplet – a grouping of 4
What is a prime quadruplet?
a set of four prime numbers in the form {p, p+2, p+6, p+8}
A representative of the closest possible grouping of four primes larger than 3
Examples
The smallest prime quadruplet is {5, 7, 11, 13} followed by {11, 13, 17, 19}
All prime quadruplets take the form {30n+11, 30n+13, 30n+17, 30n+19} with the exception of the first prime quadruplet.
The first few values of n which give prime quadruples are n=0, 3, 6, 27, 49, 62, 69, …
Properties
The width of a prime quadruplet is 8.
Three consecutive odds cannot be a part of a prime quadruplet since can interval of seven or less cannot contain more than three odd numbers unless one of them is a multiple of three.
Properties
Prime quadruplets that take the form {30n+11, 30n+13, 30n+17, 30n+19} are called prime decades.
The terms in the prime decade all start with the same number.
Some History
In 1982 a 45-digit prime quadruplet was discovered by M. A. Penk.
In 1998, the prime quadruplet with more than 1000 digits was found at the end of an 8 day search on a computer that used 1400 MHz of Pentium computer power.
Other Prime groups
There are also Prime Quintuplets keeping the same form {p, p+2, p+6, p+8} as the prime quadruplets with the addition of p-4 or p+12
and Prime Sextuplets which is when both p-4 and p+12 are prime with {p, p+2, p+6, p+8}
Big Picture Question
Are there infinitely many prime quadruplets?
References
http://www.jstor.org/discover/10.2307/3620774?uid=8366280&uid=3739840&uid=2&uid=3&uid=67&uid=62&uid=3739256&uid=8366248&sid=21102911218451
http://www.javascripter.net/math/primes/quadruplets.htm
http://mathworld.wolfram.com/PrimeQuadruplet.html
http://en.wikipedia.org/wiki/Prime_quadruplet
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