Primesassumsofirrationalnumbers
SimonPlouffe
July30,2016
IntheRamanujannotebooks,oneformulaisparticularlyremarquableis
24
1
1
Thisformulasuggeststhattherecouldbeotheridentitieswithrationalsorprimes,afterhaving
searched many forms, one in particular seems to be able to generate the prime numbers. The
Ramanujanformulaworksfor
exponents,ifweuseexponentsoftheform
then,
forexample:
691
16
1
65536
1
Here691isthenumeratorofthe12’thBernoullinumber:‐691/2730.TheRamanujanformula
iscloselyrelatedtonumeratorsoftheBernoullinumbers,oneexampleis
24
Thatnumberis
1
1563446 … 036059151, 1077
,seeA112548oftheOEISforreferences.Thebiggestknownprimeofthat
formbeing240 ∑
,a71299digitsprime.
Byexploringtheform
,IhavefoundotherprimesnotrelatedtoBernoullinumbers.
17
37
2662
251
Ifwepose
32
1
6087
1
5632
1
8192
1
1
68343
1
1
2016
4096
1
1
,
Thenwecanwriteeachprimeasasumofthe
691 ≡ 1
,
function,like
11,1 , 11,4
Disregardingthecoefficientsoftheintegerrelation.
Wehavethefollowinglistofprimes.
Prime
Linearcombination of
2
,
,
,
,
,
3
,
,
,
,
,
5
,
,
,
,
,
7
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
11
,
13
,
17
19
,
23
,
,
83
,
,
,
,
167
,
,
,
,
251
,
691
71840783209099
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
References
1 MiltonAbramowitzandIreneStegun,HandbookofMathematicalFunctionswithFormulas,
Graphs,andMathematicalTables,NewYork:DoverPublications,1972 originalin1964 .
2 StevenFinch,Centralbinomialcoefficients,
http://www.people.fas.harvard.edu/~sfinch/csolve/cbc.pdf
3 EricWeissstein,DirichletL‐series,http://mathworld.wolfram.com/DirichletL‐Series.html
4 Wikipedia:DirichletCharacter
http://en.wikipedia.org/wiki/Dirichlet_character
5 Psiorpolygammafunction,Maple18,
http://www.maplesoft.com/support/help/maple/view.aspx?path Psi
6 Espinosa,O.,andMoll,V.AGeneralizedPolygammaFunction.IntegralTransformsand
SpecialFunctions, April2004 :101‐115.
7 SimonPlouffe,TheArtofInspiredGuessing,http://www.plouffe.fr/simon/inspired.htmlalso
onthevixrasitehttp://vixra.org/author/simon_plouffeandontheArXivsite
http://arxiv.org/find/all/1/all: AND simon plouffe/0/1/0/all/0/1
8 RichardJ.Mathar,TableofDirichletL‐SeriesandPrimeZetaModuloFunctionsforSmall
Modulihttp://arxiv.org/abs/1008.2547
9 SimonPlouffe,ThemanyfacesofthePolygammafunction,http://vixra.org/abs/1603.0426