Open problems: Twin prime conjecture

Oops, so much for posting once a week! My excuse is that I’ve been hard at work on my book. Well, nothing to do but get right back at it. I promise* I will be better** about posting regularly*** from now on.

* I had my fingers crossed when I typed that. Seriously.
** For low values of ‘better’.
*** Some restrictions may apply.

Anyway, I thought I would begin an intermittent series describing some currently open (unsolved) problems in mathematics. It’s pretty fascinating that in mathematics (unlike in many other academic disciplines) it’s not too hard to come up with questions that are very easy to understand, but incredibly difficult to answer! It just goes to show that sometimes, the simplest questions end up being the deepest.

Today, I’m going to talk about the twin prime conjecture. A prime number, as you may recall, is a positive integer greater than 1 which has no divisors other than 1 and itself. For example, 17 is prime, since there is no number less than 17 that evenly divides it, but 18 is not prime (it is composite) since it is divisible by, for example, 3. The first few prime numbers are thus 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37… Prime numbers serve as building blocks for all the other integers, since any integer can be written uniquely as a product of primes (this is called the integer’s factorization).

Let’s look at the sequence of distances from each prime to the next: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, … See a pattern? I sure don’t. In fact, this is one of the most enigmatic things about the prime numbers: although there are theorems describing approximately how often primes occur, there is no neat pattern (that anyone knows of) which describes exactly when primes will occur. As you go through bigger and bigger integers, primes tend to get rarer… yet there is still no pattern. On average, the higher you go, the farther apart the primes will be — yet every now and then, primes still occur right next to each other, like 29 and 31. Pairs of primes which are only two away from each other are called twin primes. Larger examples of twin primes include 101 and 103, 281 and 283, and the whopping

2003663613 \cdot 2^{195000} \pm 1,

a gargantuan pair of twin primes with 58711 digits each! This is actually the largest known pair of twin primes, discovered just this year, on January 15. So, here’s the question: are there infinitely many pairs of twin primes, just like there are infinitely many prime numbers? Or do the primes eventually get so few and far between that the twin primes stop? No one knows! The twin prime conjecture (a conjecture is like a mathematical guess) states that there are an infinite number of twin primes — most mathematicians believe that it’s true, but no one has been able to prove it.

A challenge to conclude: we could define “triplet primes” as a set of three consecutive odd numbers p, p+2, and p+4 which are all prime. This wouldn’t be a very useful definition, however, since 3, 5, 7 is the only set of triplet primes. Why?

You can learn more about twin primes and the twin prime conjecture at MathWorld.

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3 Responses to Open problems: Twin prime conjecture

  1. Why no more triplet primes? Because you’ll always have one number among p, p + 2, and p + 4 that’s divisible by 3. Whenever you increment p by 2, the new p + 4 is the old p + 6 and the other two numbers are repeats from the previous set.

    I do take a peculiar interest in primes and was glad to enlighten Mr. Stern to the fact that 1 is technically not one. Not a prime, I mean.

    One thing I liked from the PC game “Rama” was the coding use of numbers from the sequence 41, 43, 47, 53, 61, 71… which produces 40 consecutive primes before reaching 41 squared. Supposedly, no starting prime to which you can add 2, 4, 6, etc. will yield more consecutive primes than that. I found independently that the only smaller starting primes to which you can add that pattern of differences to produce only primes until the starting prime’s square are 2, 3, 5, 11, and 17.

    That huge pair of twin primes… it seems to me that the practical value of knowing them is absolutely trivial. Whoever took the trouble to discover them must have been having fun.

  2. Brent says:

    You got it! And you’re right, one is usually considered not to be a prime, even though of course it is not divisible by anything other than itself and one — the reason being that if one is considered prime, you cannot uniquely factor numbers into products of primes — e.g. 6 could be factored as 3*2, 3*2*1, 3*2*1*1, … and so on.

    That sequence of primes comes from the prime-generating polynomial n^2 + n + 41, discovered by Euler. The fact that there isn’t a better one, as you mention, is connected to some pretty deep mathematics (which I don’t pretend to understand)! There’s more info if you follow the link.

    As for the practical value of finding large twin primes… it seems to me sort of the mathematical equivalent of, say, climbing very tall mountains. Is there an obvious, practical benefit? Not really. Can you make the argument that there are indirect benefits? Certainly. (It should be noted that the famous Pentium bug was discovered by someone calculating twin primes!) In any case, do human beings tend to take on challenging yet not obviously practical endeavors just for fun? You bet. =)

  3. vinit says:

    hello!
    yes it is indeed a surprising thing that many people spend their whole lives searching for very huge numbers like the largest twin primes or the mersenne primes. infact some people have spent a lot of time searching for the numbers in the decimal expansion of pi.
    silly as it may seem, it is not entirely without merit as most of today’s major encryption systems like the RSA etc need such huge primes for their functioning. hence it is not totally useless. but if you go after merit, what is the big merit of number theory as it is? i too am a big devotee of number theory and i find my solace in a famous saying of faradays’.
    once while queen victoria was visiting his lab, Faraday showed her some of his works in electromagnetism. remember that at that time no one even knew what magnetism was. so the queen asked him ” But what use is all this of?” to which faraday humbly replied “but madam, of what use is a baby? “

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