PRIME QUADRUPLETS Mathematics Number Theory

1. By Megan Duke – Muskingum University PRIME QUADRUPLETS MathematicsNumber Theory

2. Review • Prime – a natural number great than 1 that has no positive divisors other than 1 and itself. • Quadruplet – a grouping of 4

3. 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

4. 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, …

5. 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.

6. 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.

7. 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.

8. 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}

9. Big Picture Question • Are there infinitely many prime quadruplets?

10. 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