A previously little-known lecturer has made a significant breakthrough in the exploration of prime numbers. Dr Yitang Zhang of the University of New Hampshire has brought mathematics closer than ever before to confirming that twin primes are infinite.

A twin prime is simply two consecutive prime numbers that differ only by two. The lowest examples are three and five, five and seven, and 11 and 13. The highest known twin primes run to more than 200,000 digits.

As researchers calculate more and more prime numbers, they’ve always been able to find twin primes popping up now and again, even if they appear at wider and wider intervals. Though unproven, a widely accepted theory (the Twin Prime Conjecture) is that the number of twin primes is infinite — in other words, you can identify any twin prime you like and there’ll be one higher, even if you haven’t found it yet.

It’s a subject that interested Yitang, whose current academic posting didn’t come immediately after achieving a doctorate in 1992 — he’s said to have spent time working as an accountant and even a Subway sandwich maker while looking for a position in the US.

Last month Zitang submitted a paper on twin primes to the Annals of Mathematics, where editorial staff were so impressed they bumped it up the line for peer review. That review has now described the paper as “a landmark theorem in the distribution of prime numbers.”

The Simons Foundation has a detailed explanation of Zitang’s research and findings. He spent three years thinking about the problem, only to suddenly come up with a solution while killing time in a friend’s back yard before a night out.

The breakthrough involves the way mathematicians filter out numbers that could be primes. A 2005 technique, known as GPY, helps first identify numbers that might be part of twin primes, leaving an easier and quicker task of verifying whether this is the case in each instance.

Zitang has modified GPY in a way that means rather than just check specific numbers, he can prove a theory. In this case, he’s proven there’s an infinite number of pairs of consecutive primes that are separated by 70 million or less. That may sound incredibly vague compared to the separation of two, but the key here is providing proof.

According to Zitang, further refinement should drastically reduce the gap between primes that can be proven to repeat infinitely. However, he says it’s unlikely the system can ever prove twin primes, and instead the most likely limit is proving infinite pair separated by no more than 16.

(*Picture credit: Lisa Nugent, UNH Photographic Services*)

too much math, not enough subway.

but isnt this obvious the number system is infinite therefore there will be infinite consecutive prime numbers- all this math is lost on a biologist!

This is proposing infinite prime numbers; it is proposing infinite twin prime numbers and improved methods of finding them.

No. You can have an infinite series of numbers that adds up to a finite number. For example, 1/2 + 1/4 + 1/8 + 1/16 + … adds up to 1. This makes sense, if you think about it along the lines of Zeno’s paradox. Start with one, then take half of it away (1/2). From the remaining number, take half of that away (1/4), and so on. As long as you split the remaining number in half, you can divide 1 into infinite parts.

An infinite series can be divergent (adding up to infinity), convergent (adding up to some actual number), or conditionally convergent (adding up to infinity OR some number *depending on how you add it up*! How weird is that!?)

For our twin prime problem, as numbers get higher and higher, pairs of twin primes get further and further apart. The question is whether they spread so far apart that, even if you go to infinity, there are only a finite number or if there are actually infinitely many.

We believe there are infinitely many, but we haven’t figured out how to (or if we can) prove it yet.