LYCOS RETRIEVER 
Prime Numbers: Miscellaneous
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One of the most important conjectures in prime number theory. When (and if) it is proven, many of the bounds on prime estimates can be improved and primality proving can be simplified.
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In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. Specifically, it is one which cannot be written as the knot sum of two nontrivial knots.
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What had eluded researchers until now... was a prime-detecting method that not only always yields a correct answer but also meets another important criterion: efficiency of calculation. Instead of an algorithm requiring an amount of computer time that grows exponentially with the target number's size, computer scientists wanted one that grows more slowly, say, at a rate that's only proportional to the number size or, more likely, a straightforward power of the number size.
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Last year Connes proved that his prime-based quantum system has energy levels corresponding to all the Riemann zeros that lie on the critical line. He will win the fame and the million-dollar prize if he can make one last step: prove that there aren't any extra zeros hanging around, unaccounted for by his energy levels.
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A central problem in algebraic number theory is how a prime ideal factors when it is lifted to an extension field. For example, in the Gaussian integer example above, (2) ramifies into a prime power (1 + i and 1 − i generate the same prime ideal), prime ideals of the form (4k + 3) are inert (remain prime), and prime ideals of the form (4k + 1) split (are the product of 2 distinct prime ideals).
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[I]t is not prime. Then according to VII.31, some prime g divides it. But g cannot be any of the primes, since they all divide m and do not divide m+1.
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