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if( i == 0 ) return 1; HugeInt x = witness( a, i / 2 , n ) ; if( x = = 0 ) / / if n is recursively composite, stop return 0 ;

/ / n is not prime if we find a nontrivial square root of 1 HugeInt y = ( x * x ) 4 n; if( y == 1 & & x ! = 1 & & x ! = n - 1 ) return 0;

if( i % 2 == 1 ) y = ( a * y ) % n ; return y ; 1

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/ / Make NUM-TRIALS calls to witness to check if n is prime template <class HugeInt> boo1 isprime( const HugeInt & n )

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It became clear that moisture measurement using meters would be problematic. Because of accumulated salts in the recycled timbers, moisture contents varied widely. The history of the timbers, according to BRTW, included some that were taken from a marine structure and others that came from a building that stored agricultural chemicals. Several samples were oven dried, establishing that the average moisture content was much less than what the handheld moisture meters had indicated. Storage at the job site was critical to the moisture content of these materials. Timbers were stored at the site off the ground, but uncovered. Rain affected the top surface of the timbers. Water- lled mortise holes and areas around shakes and checks resulted in high moisture readings in various locations. Through numerous conversations between the North Carolina Department of Transportation, the timber frame subcontractor BRTW and I suggested that materials could evaluated and approved using a combination of handheld moisture readings veri ed with oven-dry tests.

const int NUM-TRIALS = 5 ; for( int counter = 0; counter i NUM-TRIALS; counter++ if ( witness( randomHugeInt( 2 , n - 2 ) , n - 1 , n ) return false; return true ; 1

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false positives /false negatives Mistakes randomly made (presumably with low probability) by some randomized algorithms that work in a fixed amount of time (p 377) Fermat's Little Theorem States that if P is prime and 0 < A < P, then A P I = I (mod P ) It is necessary but not sufficient to establish primality (p 379) full-period linear congruential generator A random number generator that has period M - l (p 369) linear congruential generator A good algorithm for generating uniform distributions (p 368) negative exponential distribution A form of distribution used to model the time between occurrences of random events Its mean equals its variance (p 373) period The length of the sequence until a number is repeated A random number generator with period P generates the same random sequence of random numbers after P iterations (p 369) permutation A permutation of 1 , 2, , N is a sequence of N integers that includes each of I , 2, , N exactly once (p 365) Poisson distribution A distribution that models the number of occurrences of a rare event (p 37 1 ) pseudorandom numbers Numbers that have many properties of random numbers Good generators of pseudorandom numbers are hard to find (p 366) random permutation A random arrangement of N items Can be generated in linear time using one random number per item (p 374) randomized algorithm An algorithm that uses random numbers rather than deterministic decisions to control branching (p 376) seed The initial value of a random number generator (p 369) trial division The simplest algorithm for primality testing It is fast for small (32-bit) numbers but cannot be used for larger numbers (p 378) uniform distribution A distribution in which all numbers in the specified range are equally likely to occur (p 367) witness to compositeness A value of A that proves that a number is not prime, using Fermat's Little Theorem (p 379).

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