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Primes Ali Salman Soutcho Toure Prof. Geoffrey Exoo Prof. Jeff Kinne

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15 computers, 4 cores each. Each core can do around 1billion operations/s. 60 cores running 24/7. SETI@home run 225,534 cores. Computer Power

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Goals ● Goal 1 ● Find large primes composed of 2digits. ● Like « 1616161661666111 » or « 15515151515511 » ● Goal 2 ● Find large primes. ● Goal 3 ● Find large Sophie Germain primes.

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Primes Example : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 … How can we check if a number is prime ? Trial Division Lucas Theorem Is 8,876,044,532,898,802,067 (19 digit) a prime ?

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Trial Division Checks if n can be divided by any number that’s greater than 1 and less than n. Example, 47 47 can only be divided by {1, 47} Example, 49 49 Can be divided by {1, 7, 49} 8,876,044,532,898,802,067 { 1500450271, 5915587277 }

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Checks if n is prime by taking the factors of n-1 and doing a sequence of tests on them. Lucas theorem Example N = 47 N-1 = 46 Factors of 46 {2, 23}

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Results 1. Finding primes that are made out of 1’s and 5’s. n = 5,551,151,151,151,111,151 (19 Digits).

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n = 511151111511111515115115555151155155511515 151515155151151511111151155151115151111151 5151115555151111 ( 100 digit ) factors for (n-1): 1: 2 2: 5 3: 29 4: 755861 5: 42189106789 6: 4363477260190361 7:1266707453760963996064804203139157071722 98481447007567082016637811 I was able to get 30-40-70 digit primes.

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Lucas Theorem Example N = 47 N-1 = 46 Factors of 46 {2, 23} 2. Finding the biggest prime What if we took random primes and multiplied them together ? (2*5)+1 = 11 (prime) (2*7)+1 = 15 (not a prime)

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Currently working on Sophie Germain prime p and 2p+1 are both prime. p = 2981625476911166341900898692545033822173452735465 0172738418743608467276189780160915299673113327070 48954384883904760278503 (121 digits) 2p+1 = 5963250953822332683801797385090067644346905470930 0345476837487216934552379560321830599346226654140 97908769767809520557006 p = 5 2p+1 = 11 5 and 11 are primes. Therefore, 5 is a Sophie Germain prime.

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Future Goal Reaching the lists of the largest prime Or Sophie Germain prime. Records 5000th largest known prime (as of now) is 287,407 digits. 20th largest Sophie Germain prime (as of now) is 29,628 digits.

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The End Students : Ali Salman Soutcho Toure Po-Ching Liu Troy Schotter Karthik Tottempudi Professors : Jeff Kinne Geoffrey Exoo

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