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Enticing Nuggets In The World of Primes

Last week I mentioned two new math discoveries, one involving finding the value of e within Pascal's triangle and the other involving a new curve based on the Golden Ratio phi.

After writing that note, another new idea (or nugget) surfaced on my desk that I need to share. As perhaps with you, the names involved caught my eye.

First, what is special about the numbers pairs: (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)...etc?

Or, what is special about these number pairs: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467)...etc?

Now, you probably are aware that the number pairs of primes differing by two are called twin primes. Examples, are (3,5), (5,7), (11,13), etc.... Plus, one of the big problems in mathematics is determining if the number of twin primes is infinite.

But, what are these other two sequences of numbers pairs? Again, they involve only pairs of primes, differing by four and six respectively.

I was surprised to learn that they have names--the "cousin primes" and the "sexy primes" respectively. Plus, both have a rich mathematical history, including the consideration as to whether there are an infinite number of each type.

Just in case you are wondering, no mystical meaning underlies the term "sexy prime." Rather, it is traced back to the fact that the Latin word for "six" is "sex."

In addition to the sexy prime pairs, there are sexy prime triplets such as (7,13,19) and sexy prime quadruplets such as (5,11,17,23). And it is easy to prove not only that (5,11,17,23,29) is the only sexy prime quintuplet, but also that no longer sequence of sexy primes is possible. Give it a try.

Who thinks up all of these interesting things? For example, I have known about twin primes since I was in high school, yet never thought of considering differences of four or six. Why not try a difference of eight...as I could find no explorations of this extension. The first example is 89 and 97...and then....

I should note that mathematicians Goldston, Pintz, and Yildirim showed in 2005 that there exist pairs of prime numbers which are very close together. By assuming something known as the Elliott-Halberstam conjecture, they proved that there are infinitely often primes differing by 16 or less.

Number theory and its world of primes offer so many neat ideas. If you have time, explore these special ideas: Lucky primes, fortunate primes, strong primes, good primes, illegal primes, happy primes, safe primes, balanced primes, titanic primes, gigantic primes, mega primes, pseudo-primes, almost primes, interprimes, probable primes, industrial-grade primes...and the list goes on....perhaps the list is infinite as well?