Re: using the prime gaps to make a convergent series

*To*: mathgroup at smc.vnet.net*Subject*: [mg52167] Re: using the prime gaps to make a convergent series*From*: Roger Bagula <tftn at earthlink.net>*Date*: Sat, 13 Nov 2004 04:40:17 -0500 (EST)*References*: <cmklnp$jl0$1@smc.vnet.net>*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

I reasoned that there might also be 2^n*F(n)+1 type primes as a result of the gaps. It appears that there are! (* prime gaps as a product function to get primes*) w[n_]=Prime[n+1]-Prime[n] p[n_]=1+Product[w[m],{m,2,n}] digits=60 a=Delete[Union[Table[If[PrimeQ[p[n]]==True,p[n],0],{n,1,digits}]],1] {2,3,5,17,257,12289,14155777,169869313,4076863489,32318253138475745281, 12806790724213976503626296721408001,307362977381135436087031121313792001} Roger Bagula wrote: >This series works as a sum because the Prime gaps are in general >a factor of two: >Prime[n]=Prime[n-1]+Gap[n] >Gap[n]=2*w[n] >Product[1/Gap[n],{n,1,Infinity]=0 as 1/2^n->0 >In general except for the first value w[n] behaves as a chaotic >with minimum 1 and a building maximum on a cycle. >The study of what are called prime pairs ( primes 2 apart by gap) >shows this cyclic building behavior and is well known. >The cycle maximum is thought to approach infinity in a countable manner: >wmax[m]=wmax[m-1]+1 >where > m=f[Prime[n]] > > >(*Product converges to limit of zero as 1/2^n*) >f[m_]=Product[1/(Prime[n+1]-Prime[n]),{n,1,m}] >(* number as sum of Product gap function increments*) >Digits=200;a=Table[f[n],{n,1,Digits}]; >b=N[Apply[Plus,a],Digits] >(* digits of the new irrational number*) >c=Table[Floor[Mod[b*10^n,10]],{n,0,Digits-1}] > >{1,8,5,6,7,0,8,6,1,6,2,9,0,1,3,6,0,9,9,0,8,3,9,6,6,7,8,9,5,1,2,4,5,2,2,5,1,3, > >8,4,6,0,3,2,7,7,1,6,1,1,9,5,9,8,2,7,9,4,8,1,8,8,6,0,8,6,7,8,6,0,4,5,0,0,8,6, > >7,1,6,9,6,1,3,2,2,1,9,0,7,4,6,2,7,2,8,3,4,7,1,2,5,6,5,4,9,5,2,5,4,3,6,4,3,0, > >2,0,8,1,1,4,0,1,6,1,8,4,9,1,6,0,7,5,1,7,6,7,3,9,4,3,1,0,4,5,2,0,8,2,1,3,6,7, > >6,5,6,7,3,4,5,7,8,4,7,6,2,6,3,5,7,8,1,3,4,1,6,3,7,5,2,4,9,4,3,8,9,9,1,5,4,8, > 6,1,3,6,4,3,3,1,6,2} >Respectfully, Roger L. Bagula > >tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : >alternative email: rlbtftn at netscape.net >URL : http://home.earthlink.net/~tftn > > > -- Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn at netscape.net URL : http://home.earthlink.net/~tftn