Re: using the prime gaps to make a convergent series

• To: mathgroup at smc.vnet.net
• Subject: [mg52034] Re: [mg52021] using the prime gaps to make a convergent series
• From: DrBob <drbob at bigfoot.com>
• Date: Mon, 8 Nov 2004 03:13:13 -0500 (EST)
• References: <200411070604.BAA18080@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```My apologies for misleading you in our recent discussions of the gaps between consecutive primes. I'd forgotten to really think about those gaps!

There's a well-known conjecture (unproven either way) that there are infinitely many twin-prime pairs, where p and p+2 are both prime. The first book I picked up listed this pair, for instance, both of which are known to be prime:

2409110779845*2^60000 - 1
2409110779845*2^60000 + 1

Look here for a much bigger pair:

http://www.fact-index.com/t/tw/twin_prime.html

The same book says the sum of reciprocals of all the twin-primes is finite. It's called the Brun constant, and it's between 1.82 and 2.15 -- probably about 1.9021605822. It sounds odd to say it that way, but that's how much we know, or how much was known when Crandall and Pomerance's book was published (Prime Numbers: A Computational Perspective).

The gap between twin-primes is 2, so if the conjecture is true (infinitely many twin-primes), the gap does not tend to infinity. Yet the gaps aren't bounded, either. The gaps series contains larger and larger numbers interspersed with small numbers. For 100,000 gaps, here are the maximum gap and the number of twin-primes found:

Clear@gap
gap[n_Integer]/;n>0:=Prime[n+1]-Prime[n]
gaps=Array[gap,{10^5}];
Max@gaps
Count[gaps,2]

114
10250

Aside from the twin-prime conjecture, it's known that PrimePi[x] is asymptotic to x/Log[x]. Differentiating, we get an approximation to the number of primes per unit increase in x:

Simplify[D[x/Log[x], x]]
(-1 + Log[x])/Log[x]^2

The inverse of that,

Simplify[1/D[x/Log[x], x]]
Log[x]^2/(-1 + Log[x])

is the approximate gap to the next prime. (That's a loosely derived _average_ in the vicinity of x, since it's based on an _asyptotic_ approximation to PrimePi, not one that's accurate at every x. We only know the ratio of PrimePi[x] to x/Log[x] tends to one as x gets large.) This approximation says average gaps do get large:

Limit[Log[x]^2/(-1 + Log[x]), x -> Infinity]
Infinity

Looking at a few plots like this:

Plot[Log[x]^2/(-1 + Log[x]), {x, 10, 10^100}]

shows the gaps tend to infinity VERY slowly.

Here's a comparison of actual gap averages with the approximation. (None of this is intended to be precise; averaging gaps "in the vicinity" of x is vague at best. I took 5 gaps on either side of the nearest prime no bigger than x.)

average[x_Integer] := N[Module[{n = PrimePi[x]},
{(Prime[n + 5] - Prime[n - 5])/10,
Log[x]^2/(-1 + Log[x])}]]
average /@ (10^Range[2, 12])

{{4.2, 5.88255}, {6.4, 8.07702}, {
13.2, 10.3321}, {13.4, 12.608}, {15., 14.8935}, {20.8, 17.1842}, {
22.6, 19.4781}, {32.6, 21.774}, {26.4, 24.0713}, {21.8, 26.3695}, {24.4, \
28.6686}}

Bobby

On Sun, 7 Nov 2004 01:04:00 -0500 (EST), Roger Bagula <tftn at earthlink.net> 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
>
>
>
>

--
DrBob at bigfoot.com
www.eclecticdreams.net

```

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