using the prime gaps to make a convergent series

• To: mathgroup at smc.vnet.net
• Subject: [mg52021] using the prime gaps to make a convergent series
• From: Roger Bagula <tftn at earthlink.net>
• Date: Sun, 7 Nov 2004 01:04:00 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```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