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Twin Prime Conjecture
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One way to approach the twin prime problem is to look at the gaps between successive primes. For example, if p(1), p(2), p(3), . . . denotes the sequence of all primes, are there infinitely many values of n for which p(n + 1) – p(n) is less than 10, say, or less than 100? If you can find a gap for which there are infinitely many pairs of successive primes that differ by no more than that gap, maybe you can start to bring the gap down. If you get the gap down to 2, you will have proved the twin prime conjecture.
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Brilliant mathematician Linda Ruether has spent her life trying to unlock the age-old mathematical mystery of the Twin Prime Conjecture. She has sacrificed everything but this tantalizing dream has remained just out of reach. When an eighteen-year-old genius stumbles upon the solution, she can't resist the temptation to gain her own intellectual immortality, whatever the cost. She might just succeed. She might be able to join the pantheon of giants. Or is there something that she has overlooked?
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Kelly and Pilling determined all prime separations between pairs of twins in various ranges selected from integers between 79,561 and 4,020,634,603. They then determined the relative frequency of occurrence of each separation in a given range. They observed that, for all sufficiently large ranges, the relative frequencies appear to obey a surprisingly simple logarithmic relationship.
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This conjecture would imply the twin prime and Sophie Germain conjectures, as well as the Green-Tao theorem; it ... implies the Hardy-Littlewood prime tuples conjecture as a special case. There is a quantitative version of this conjecture which predicts a more precise count as to how many solutions there are in a given range, and which would then also imply Vinogradov’s theorem, as well as Goldbach’s conjecture (for sufficiently large N); see this paper for further discussion. As one can imagine, this conjecture is still largely unsolved, however there are many important special cases that have now been established - several of which via the Hardy-Littlewood circle method.
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These collections of data are of great interest to investigators examining the distribution of twin primes among all primes and the gaps between consecutive twins. The data show that, like primes, twin primes tend to become more scarce as their numerical value increases.
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Despite hundreds of years of effort, no one has proved the twin prime conjecture. The closest anyone has come was in 1966 when Chinese mathematician Chen Jingrun found that there are infinitely many primes p such that p + 2 is either prime or a product of two primes.
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Twin Prime Conjecture