SEXY PRIMES

Before you begin to think that this page of WTM is going to be "x-rated", we urge you to put that thought out of your mind right now. You see, we have merely created a new category for classifying numbers. (We assure you that we're not going to talk about boy-numbers and girl-numbers.) Rather we will do this with that very important kind of number, the prime number.

But first a little review of well-known topic in basic number theory.

## "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.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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

Now we're ready to explain what is "sexy" about all this. Notice that sometimes in two consecutive rows two primes appear "one above or below another." Find 13 and 19 to see what we mean. And the positive difference of the two primes is (you've guessed it!) 6. If you recall your Latin number words -- no, we didn't say Roman numerals --, "sex" is Latin for "six." That's why we have the name of sextillion in our work with large numbers; and sextuplets for the birth of 6 babies at once.

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:

Let's return for a moment to the matter of twin primes. What is the strongest statement that we can say about the set of numbers that lie between twin primes? Oh, it's easy to say that they are all even numbers, but that's not very exciting, is it? Let's go for more!

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?