Nearly final piece on twin primes…


One more post on Prof Tao’s blog today, recorded here for posterity:

——————-

So let’s start with the assumption that the claimed definitions of Nn and Tn are proven. I think I’ve proven the first, and can only say I believe the second from numerical experimentation, and will try to defend it, given more time and more miles.🙂

Let’s pick up from two posts back with the idea that the ratio of the density of twin pairs within the set of primes to the density of primes within the set of natural numbers can be expressed as:

D(t:p)/D(p:nat) = [Tn/Nn]/[Nn/Pn#] = Tn(Pn#)/(Nn)^2, with n approaching infinity

Where D(t:p) means the density of twin pairs within the set of primes, and D(p:nat) = density of primes within the set of natural numbers.

This can be re-written as:

D(t:p)/D(p:nat) = product for i = 1 to infinity {Pi x P(i – 2)/[P(i-1)]^2} ——- eqn [2]

Where we include P0 = 1 to allow us to simplify the examination of the products with no change to the result.

This can be written as:

D(t:p)/D(p:nat) = limit as n approaches infinity {Pn x P(n – 2)/[P(n-1)]^2} x … x {13 x 11/12^2} x … x {3 x 1/2^2} x {2 x 1/1^2}

Numerical experimentation shows this series converges quite nicely to 1.320… as claimed. I won’t attempt to work out the arithmetic, but by inspection, as n increases, the nth term steadily converges on 1, being always < 1, and with n = 1, 2, 3 and 4 we get 2, 1.5, 1.406…, and 1.367…

Without knowing the actual value of this infinite product, we can obtain a lower bound by examining:

Product for i = 1 to infinity {i x (i – 2)/(i – 1)^2}

which converges on 1.

This latter product includes all the terms in eqn [2], but also includes additional terms less than 1 for values of i that are not prime. Hence it decreases more quickly than eqn [2], and is always lower than eqn [2], after the first two terms, when the same number of terms has been included in both products. Therefore in the limit, eqn [2] converges on a positive number that is less than 1.367… (by inspection) and not less than 1.

So, if we believe the definition of Tn as claimed, I think we have shown that the set of twin prime pairs is infinite, since its density within the infinite set of primes is at least equal to the density of primes within the infinite set of natural numbers.

Of course the definition of Tn is not yet proven. A few more miles to be run first…🙂

Questions, comments, rotten tomatoes welcome.

Hopefully no fatal typos in this post, fingers crossed…

About petequinn

I'm a Canadian geotechnical engineer specializing in the study of landslides. I started this page to discuss some mathematical topics that interest me, initially this involved mostly prime numbers, but more recently I've diverted focus back to a number of topics of interest in geotechnique, geographic information systems and risk. I completed undergraduate training in engineering physics at Royal Military College (Kingston, Ontario), did a masters degree in civil (geotechnical) engineering at University of British Columbia (Vancouver), and doctorate in geological engineering at Queen's University (Kingston). I was a military engineer for several years at the beginning of my career, and did design and construction work across Canada and abroad. I've worked a few years for the federal government managing large environmental clean up projects in Canada's arctic, and I've worked across Canada, on both coasts and in the middle, as a consulting geotechnical engineer. My work has taken me everywhere in Canada's north, to most major Canadian cities and many small Canadian towns, and to Alaska, Chile, Bermuda, the Caribbean, Germany, Norway, Sweden, Bosnia, and Croatia. My main "hobby" is competitive distance running, which I may write about in future.
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