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>
• 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
>
>alternative email: rlbtftn at netscape.net
>
>
>

--
Respectfully, Roger L. Bagula