Goals

• 5000th largestknownprime• Mustbeabout300,000digits

• 20th largestSophieGermain prime• Mustbeabout30,000digits

PrimeNumber

• Canonlybedividedbyitselfandone.

• Examples: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47…

Results

• Aftervariousimprovementstocodeefficiency…

• Weareabout½waythroughthesearchfor30,000digitSophieGermain primeafter2weeksofcomputertime

• Searchingfor300,000digitprimewilltakeanother2weeks ofcomputertime

ComputersUsed

About220cores:• 20dual-core3.2Gzi5machines,• 30quad-core3.1Gzi5machines,• 15quad-core2.8Gzi5machines.

PrimeNumberTest:TrialDivision

•Divisionofnbyasequenceofnumbergreaterthan1andlessthann.•Helpseliminateunwantednumbers

Example,47

47canonlybedividedby{1,47}

Example,49

49Canbedividedby{1,7,49}

PrimeNumberTest:FermatTest

•an-1 modn=1•Ifthenumberpassesthetestthenitmightbeaprime,butifitdoesnot,thenitisnotaprime.•Example:

n=4 n=5a=3 a=333 mod4 34 mod527mod4=3 81mod5=1Failed Passed

PrimeNumberTest:LucasTest

•Findprimefactorsofn.•Runasequenceoftest(almostsimilartotheFermatequation)onthem.•Example:

n=47n– 1=46Factors:{2,23}

PrimeNumberTest:LucasTest

Howrareareprimenumbers ?

𝝅(𝒏) – #ofprimesupto𝒏

𝝅 𝒏 ∼ 𝒏

𝒍𝒏 𝒏

1digitnumbers:4 areprime2digitnumbers:21 areprime30,000digitnumbers:about1/70,000 areprime300,000digit numbers:about1/700,000 areprime

Sohpie Germain Primes

• 𝑛 and2𝑛 + 1 arebothprime

• 30,000digitnumbers• ∼ ( 8

9:,:::)< areSophieGermain primes

HowlongdoesittaketogetaprimeandaSG

•PrimeNumber300,000digits•9.8× 10^149986 years(TrialDivision)•2695 years(FermatTest)

•SophieGermaine30,000digits•6976years(TrialDivision)•98 days(FermatTest)

Methodused

Isnprime?Loop(){1- Trialdivisiontryn/2

n/3n/5…..n/9973

2- Fermat'stest:𝒂𝒏>𝟏𝒎𝒐𝒅𝒏 = 𝟏3- Lucastheorem}