Built on Facts

Sunday Function

Posted by Matt Springer on September 28, 2009
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Sometimes in math we’ll understand one aspect of a problem very well, while at the same time we understand another aspect of a problem very poorly. For instance, take the prime numbers. According to the prime number theorem, the number of prime numbers below x is approximately given by:

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i-7b7142f1efa09b47c3b8242f89bcb9f4-graph.png

Where pi(x) is the prime counting function and ln(x) is the natural logarithm. As you keep counting your way up the number line, you’ll encounter more and more primes. They thin out and become more and more rare, but nonetheless there’s an infinite number of them as you keep going. The number that you encounter is roughly x/ln(x), and that approximation gets better and better as a percentage as you go up.

Following from this, the density of prime numbers near x will be about 1/ln(x) and the nth prine number will be approximately n*ln(n). Let’s take a look at a quick example. If x = 1,000,000, we should expect that there are about 1,000,000/ln(1,000,000) = 72,382 prime numbers below a million. In fact there are 78,498. Not too bad.

We’d also expect the millionth prime number to have a magnitude of roughly 1,000,000 * ln(1,000,000) = 13,815,501. It’s actually 15,485,863, which is within a few percent error.

Now that’s just the tip of the iceberg in terms of what we know about the prime numbers. Literally millions of pages have been written about the prime numbers and their relationship with everything from complex analysis to group theory to encryption.

But other things are a blank mystery. Consider pairs of prime numbers, where you have a number p and its twin p + 2 and both are prime. These are the “twin primes” and the first few are (3, 5), (5, 7), (11, 13), and (17, 19).

The function that counts them in analogy to the prime number theorem above is our Sunday Function. I’d write it down for you except I have no idea what it is. Neither does anyone else. We don’t even know if there’s an infinite number of twin primes, for all we know the “twin prime counting function” might flatline somewhere out along the number line because there just aren’t any more of them. Who knows?

Now assuming some statistical properties of the prime numbers, it’s possible to make educated guesses about the distribution of the twin primes. Guesses in math don’t count for much no matter how plausible they are, but it’s at least a place to start. And by plausibly guessing, mathematicians have showed some good reasons to think that the twin prime counting function behaves roughly as x / (ln(x))^2. It’s provably not any bigger than that in the sense of O notation, anyway. But that certainly doesn’t exclude an O(1) (in the notation) flatline because there’s only a finite number of twin primes.

If you can actually figure out what the Sunday Function is explicitly, you’ll be a math celebrity and you just might get yourself a Fields Medal (there’s no Nobel for math for some reason). Give it a try!

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Comments

  1. #1 Nomen Nescio
    September 28, 2009

    i’ve heard the usual urban legend about how Alfred Nobel knew one mathematician, and thoroughly hated him, so decided to leave maths out of his bequest. snopes claims it’s false, but doesn’t really give any better reason either.

  2. #2 Rob
    September 28, 2009

    I’d find the function but I’m 55 years of age and thus too old to win the Fields Medal, so why bother?

  3. #3 Joshua Zelinsky
    September 29, 2009

    There are explicit forms for Pi(x). They just aren’t interesting or take forever to use to count or some other problem.There are a number of different formula involving sums over zeros of the Riemann zeta function for example. Unfortunately, there are infinitely many terms.

    Note also that since PRIMES is now in P, part of the computational interest in an exact formula for Pi(x) is not as great as it once was. However, the known complexity of calculating Pi(x) is still very bad, even if one restricts oneself to inputing k and n and asking “Is Pi(n)=k?”

  4. #4 Dave Child
    September 29, 2009

    Can’t you demonstrate there are an infinite number of pairs the same way you can demonstrate there are an infinite number of primes? Multiple all known primes and add one and you have a new prime number. But if you multiple all known primes and subtract one, you also have a prime number, as the new number can have no prime factors.

  5. #5 Dave Child
    September 29, 2009

    Bah, never mind – just reading up on primes and saw that that product-of-primes-plus-one idea is different to how I stated it, but the same should still apply – either p-1 is prime or it has a new prime factor outside of the finite set of primes multipled to get it.

  6. #6 Dave Child
    September 29, 2009

    Bah, again. There’s no guarantee that p+1 and p-1 are both prime. just that they are prime or have a new prime factor outside the original set. I’m going for coffee.

  7. #7 JR
    September 29, 2009

    “snopes claims it’s false, but doesn’t really give any better reason either.”

    Nobel seems to have liked useful sciences. It is quite possible that he didn’t consider math one and that is the reason.

  8. #8 Janne
    September 29, 2009

    Nobel was aiming for human betterment. So physics and chemistry improve the lives of people by the way they let us control nature; medicine improves the lives of people directly; literature takes another approach to improving people’s lives and peace is of course yet another tack on the same theme. Economy is of course not an actual Nobel price at all; the Swedish Riksbank managed quite a PR-coup by getting their price to be handed out at the same time as the Nobels.

    Math is abstract and not in the business of helping people or societies, while fields like biology and geology was mostly naturalism at the time and didn’t have the kind of impact it does today (the parts that did have an impact could easily fit under chemistry and medicine).