"TWIN" PRIMES

     As all good math students know, a pair of twin primes is simply two prime numbers that have a positive difference of 2. For those who have forgotten this fact, a couple of examples should suffice to jog their memories: {11, 13} and {29, 31}.

     There are many, many pairs of twin primes out there in that big ocean of numbers. Why not go fishing for some of them...

     Now something not so well known will be demonstrated here by way of introducing a new definition for categorizing prime numbers.

     First, we begin by writing out some of the natural numbers in rows of six numbers.

     If one observes this array of numbers carefully, it will be clear that prime numbers only appear in the first and fifth columns -- that is, after we get past the two primes of 2 and 3. In fact, if you consider the math involved, it should be obvious. The second, fourth, and sixth columns are composed of only even numbers; so that takes care of them in short order. (Why can we eliminate the 3rd column almost as easily?)

     However, just because a number is in the 1st or 5th columns doesn't mean that it is prime. All we're saying is that if a number is a prime, it can only be found in one of those two columns.

"SEXY" PRIMES

     So, there you have it. If a pair of primes has a positive difference of 6, we here at WTM have declared that such primes shall henceforth and forevermore be called SEXY PRIMES. How many sexy primes can you find?

     Of course we should not limit ourselves to just two primes at a time. We can have sets of 3, 4 or even 5 primes that are sexy. These groups come to mind: {31, 37, 43}, {251, 257, 263, 269}, and {5, 11, 17, 23, 29}.

     After writing the above information and publishing it in my November 1997 issue of "Trotter Math" News, I received the following e-mail message from Monte Zerger, a math professor from Adams State College in Colorado:

     I find your "sexy numbers" fascinating and felt compelled to investigate them a bit. Here are some things you may have already discovered.

1. It is impossible to have more than four consecutive sexy primes, except for the 5, 11, 17, 23, 29 you mention. This is because the unit's digit of numbers of the form 6n - 1 or 6n + 1 will cycle through the odd digits, that cycle being either 7, 3, 9, 5, 1, 7, 3, ... in the case of 6n + 1 numbers or 5, 1, 7, 3, 9, 5, 1, ... in the case of 6n - 1 numbers. Thus every fifth number of the form 6n - 1 or 6n + 1 is divisible by 5.
   
   
2. From this it is easy to see that a string of four consecutive sexy primes must begin with a prime whose unit's digit is 1.
   
   
3. For lack of a better idea at the moment, let's call such a string of four sexy primes a "sexy foursome."
   
   
4. Looking at years, the last sexy foursome was (1741, 1747, 1753, 1759) and the next one won't be until (3301, 3307, 3313, 3319).
   
   
5. However there's something rather sexy about this century. It began with a sexy threesome (1901, 1907, 1913) and ends with another (1987, 1993, 1999). There were no other sexy threesomes in this century.
   
   

Footnote:

     There is one set of 3 primes that might be called "triplets", because they have two differences of 2 among them. We're thinking of {3, 5, 7}, of course. Can you prove that these three primes are the only ones to have this property?

     Finally, as we saw above, there was a set of sexy primes made up of 5 primes. Can you prove that there could never be a set of 6 primes, coming from 6 consecutive rows and in the same column?