2. Basic ideas
Every positive integer other than 1 can be written in a unique way as a product of primes.
(This photograph was made available with the kind permission of Harald Hanche-Olsen)
His work makes more specific the earlier family of problems raised by Alphonse de Polignac, who raised the question in 1849 if, for any even integer s, there are infinitely many pairs of primes that differ by s? The remarkable answer to Brun's question about twin primes is that no one knows. For the primes, if one sums the series whose terms are 1/p where p is a prime, the sum turns out not to be a finite number. Perhaps surprisingly, if one sums the series whose terms are 1/s and 1/(s+2) where s and s+2 are both primes, Brun showed the result is a finite number. To use slightly more technical terminology, the first series diverges and the second converges to the constant 1.9021605... , regardless of whether it has a finite number of terms (which as I noted, is still not known). The sum of the series of twin prime pairs is now known as Brun's constant. Finding its exact value has served as a challenge for mathematicians interested in computation. Thomas Nicely, during his effort in 1995 to find a more precise value for Brun's constant, discovered a flaw in the design of one of Intel's Pentium series of chips! Who would have thought that trying to solve an arcane mathematics problem would save many individuals from improperly relying on calculations done by a microprocessor? Nicely is still involved with trying to find more exact values of Brun's constant.
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