lecture15

lecture15 - Discrete Mathematics for Computer Science...

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Discrete Mathematics for Computer Science COMPSCI 102       Duke University Randomness and Computation:  Some Prime Examples
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Earth has huge file X that she transferred to  Moon. Moon gets Y. Earth: X Earth: X Moon: Y Did you get that file ok? Was the  Did you get that file ok? Was the  transmission accurate? transmission accurate? Uh, yeah.
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Gauss Let  π (n) be the number of  primes between 1 and n.  I wonder how fast  π (n)  grows?  Conjecture [1790s]:  ( ) lim 1 / ln n n n n π = Legendre
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Their estimates x pi(x) Gauss' Li Legendre x/((lnx )- 1) 1000 168 178 172 169 10000 1229 1246 1231 1218 100000 9592 9630 9588 9512 1000000 78498 78628 78534 78030 10000000 664579 664918 665138 661459 100000000 5761455 5762209 5769341 5740304 1000000000 50847534 50849235 50917519 50701542 10000000000 455052511 455055614 455743004 454011971
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J-S Hadamard Two independent  proofs of  the  Prime Density Theorem  [1896]: ( ) lim 1 / ln n n n n π = De la Vallée Poussin
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The Prime Density Theorem This theorem remains one of the celebrated achievements  of number theory.  In fact, an even sharper conjecture  remains one of the  great open problems of mathematics!
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Riemann The Riemann Hypothesis  [1859]  ( ) / ln lim 0 n n n n n π - =
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Slightly easier to show   π (n)/n   1/(2 log n) (We won’t prove it here.)
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Random  (log n) -bit number is a 
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This document was uploaded on 01/17/2012.

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lecture15 - Discrete Mathematics for Computer Science...

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