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Mathematicians Hunt the Voids Between Massive Prime Numbers

Proving prime gaps.
​Image: Christopher Sessums/Flickr

​Characterizing prime numbers is manual labor. The rules, the few rules that exist for primes, are fuzzy and uncomfortably arbitrary. Given one prime number, there's no definitive way to get the next prime number; alternatively, to test if a number is in fact prime, we're forced to just check every other smaller number to see if it divides the number in question (OK, not quite every other smaller number).

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There are patterns within the prime numbers, however. Relationships recur, as do forms. There are even certain types of primes, like the Mersenne primes, which are defined by a form where each one is one less than a power of two. Or: 2n-1.

A lot of prime number curiosity has to do with prime gaps. These gaps are just the intervals between successive primes, which have rules and patterns of their own. It's an area of research that's been stagnant/stymied for decades, resurrected only last year by a relatively out-of-nowhere mathematician named Yitang Zhang.

​In a much-heralded paper, Zhang offered a new answer/advance to the so-called twin-primes problem, setting off a flurry of new activity culminating in the ​recent release of a pair of papers​one from math big-name Terence Tao, offering proofs of the famous Erdős conjecture, which puts an upper limit on those gaps.

The general problem is of how far apart consecutive primes are "allowed" to get. Among smaller numbers, the gaps between primes are pretty small usually. Here are the first 30, making up a surprisingly tidy sequence:

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14

The gaps start getting really big soon enough. The largest known gap between two successive primes as of 2012 was 66520. The question is of how big these gaps can get. There is a limit it seems, or there should be. For many decades, the answer was left at an equation given by the Scottish mathematician Robert Rankin, which is this, but don't worry about it too much:

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The thing to note is the c, which is just a constant number multiplying all of the other junk (read up on logarithms ​here).

Paul Erdős wanted to prove a really big prime gap, where we could replace the constant at the beginning of the thing/formula above with any arbitrarily huge number. The gap could be really big then, but it went unproven until this year when Tao and co. offered their solutions, describing c as a function that could grow arbitrarily large as the sequence of natural numbers (counting numbers) grows to infinity (so long as it stays below the upper limit of the sequence).

The solution rests on a clever way of devising lists of non-prime or composite numbers using the ​factorial function (where successive numbers are multiplied together, so 3 factorial or 3! is 1 x 2 x 3 = 6). Using factorials, it's possible to build huge composite numbers (of the sort that live between primes) really quickly and all of this points to maybe being able to prove an even larger prime gap than previously assumed.

All of the giant math brains named above are now working together on the problem, and they hope to have another new-and-improved paper out this month. The result, ​Tao told Quanta Magazine, will push the above mess courtesy of Rankin to its limit, which may be very, very big.

This is actually important, at least in one respect. Cryptography depends on hunting for very large prime numbers and, if it turns out that the maximum possible gap is unexpectedly immense, it might create a hole of sorts where a prime-hunting algorithm falls into a huge span of non-prime composite numbers, which may well result in a crash or at least a bit-fart of some sort.

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It's also just kind of interesting (or really interesting, depending) to see limits put on prime numbers, which are otherwise the most devious numbers.

Here is how the mathematician ​Don Zagier put it in a 1975 lecture:

There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.

The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.

Organization finds a way.