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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. run 225,534 cores. Computer Power

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Goals ● Goal 1 ● Find large primes composed of 2digits. ● Like « » or « » ● 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, can only be divided by {1, 47} Example, Can be divided by {1, 7, 49} 8,876,044,532,898,802,067 { , }

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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 = ( 100 digit ) factors for (n-1): 1: 2 2: 5 3: 29 4: : : : I was able to get 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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( around 4,200 digits ).

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Currently working on Sophie Germain prime p and 2p+1 are both prime. p = (121 digits) 2p+1 = 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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