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It is Conjecture d that there are Infinite ly many Sophie Germain primes, but like the Twin Prime Conjecture , this has not been Proven . The first few Sophie Germain primes are : : 2 , 3 , 5 , 11 , 23 , 29 , 41 , 53 , 83 , 89 , 113 , 131 , 173 , 179 , 191 , 233, ... Currently, the largest known Sophie Germain prime is 7068555 × 2121301 - 1, discovered by Predrag Minovic in January 2005, using TwinGen and LLR . A Heuristic estimate (due to G. H. Hardy and J. E. Littlewood ) for the Number of Sophie Germain primes less than ''n'' is 2''C''2 ''n'' / ( Ln ''n'')2 where ''C''2 is the Twin Prime Constant , approximately 0.660161. For ''n'' = 104, this estimate predicts 156 Sophie Germain primes, which has a 20% error compared to the exact value of 190 above. For ''n'' = 107, the estimate predicts 50822, which is still 10% off from the exact value of 56032. A sequence {''p'', 2''p'' + 1, 2(2''p'' + 1) + 1, ...} of Sophie Germain primes is called a Cunningham Chain of the first kind. Every term of such a sequence except the first and last is both a Sophie Germain prime and a Safe Prime . If a Sophie Germain prime ''p'' is congruent to 3 mod 4, then its matching safe prime 2''p'' + 1 will be a divisor of the Mersenne Number 2''p'' - 1. Sophie Germain primes were the subject of the Eponym ous proof in the stage play '' Proof '' and the subsequent film '' Proof ''. APPLICATION IN RANDOM NUMBER GENERATION Sophie Germain primes have a practical application in the generation of Random Numbers . The decimal expansion of reciprocal 1/q will produce a Repeating Stream of pseudo random numbers of length q - 1 if q is such that q = 2S + 1 where S is a Sophie Germain Prime , such that both S and 2S + 1 are prime, with S being of the form 3, 9 or 11 mod 20. Thus “suitable” prime numbers q are 7, 23, 47, 59, 167, 179, etc (corresponding to S = 3, 11, 23, 29, 83, 89, etc.). The result is a stream of length q-1 digits (including leading zeros). So, for example, using q = 23 generates the random digits 0,4,3,4,7,8,2,6.....3,9,1,3 EXTERNAL REFERENCES
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