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Prime numbers – those famous integers only divisible by themselves and 1 – are the constituent building blocks of mathematics. Take any number, and it can be expressed as a unique product of prime numbers, a fact so important, it is known as the ‘Fundamental Theorem of Arithmetic’. Despite this, we know relatively little about the primes. Though we know that there are infinitely many of them, our inability to predict their locations on the number line is a constant source of scholastic chagrin and embarrassment.

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A stained glass window of primes…

Their impenetrable randomness has thrown up a vast sea of seemingly simple hypotheses that have proved remarkably difficult to… well, prove. A famous example is the so-called ‘Twin Prime Conjecture’, the theory that  there are infinitely many ‘Twin Primes’ (i.e. two prime numbers that have a difference of two, such as 3 and 5, 17 and 19, and 137 and 139). Mathematicians are almost certain that this statement is true. The largest known twin primes are each 200700 digits long, two mind-bendingly large numbers (for perspective, the number of atoms in the observable universe is a mere 82 digits long). This, however, does not constitute a proof that there are infinitely many – the 200700 digit monsters may well be the largest.

Nigh on no progress has been made proving the Twin Primes hypothesis since it’s initial proposal in 1847, so the news on 17th April 2013 that the Chinese-American mathematician Zhang Yitang, based at the University of New Hampshire, had made a breakthrough sent shockwaves through the mathematical community. Zhang had proved a ‘weak form’ of the Twin Prime conjecture, that rather than there existing infinitely many primes with a difference of 2, there irrefutably exists an infinite number of primes with a difference of some as yet undeterminable number… less than 70 million. Admittedly, a difference of 70 million does sound fabulously wide of the mark given that we’re aiming for a difference of 2, though it’s an excellent improvement upon the previous upper bound of infinity. Not bad for an unknown mathematician who used to pay the bills by working in his local Subway sandwich shop.

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