Three and five. Five and seven. Eleven and thirteen. Prime numbers often appear as twins, only two apart. For hundreds of years, mathematicians have wondered – is there a biggest pair of twin primes, or does the list of twins keep going forever?
A lot of small primes are also twin primes. But as you look at bigger and bigger primes, the average gap between them also gets bigger. Eventually, prime numbers are one-in-a-million, one-in-a-billion, or even less common.
However there’s also evidence for endless twin primes. Although they can be hard to find, mathematicians have found some truly large prime twins. The largest pair of twin primes currently known has over 200 000 digits in each number. These numbers are so large that mathematicians write them as a calculation, rather than a number.
We don’t know how many twin primes exist. But recently, Chinese mathematician Yitang Zhang made a very big step towards an answer. His work looks at more distant pairings. In Yitang’s work, two primes are paired if they differ by less than 70 million – so twin primes are included, but so are primes that differ by 3 347 642, for example. Yitang’s proof shows there is an infinite number of these more distant pairings.
You might think that such distant pairs are a long way from twins differing by two. But number theorists are excited by this discovery. Yitang’s result strongly suggests that the list of twin primes goes on forever. Several mathematicians are already improving the proof to make the pairs closer together. Things are looking good for an endless list of twins.
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