Have been plugging away some more looking at patterns that develop in the candidate primes and k-tuples with progressive sieving to the n-th prime. Thought I would mention a couple of interesting observations.

The first is simply an observation of behaviour that may or may not extend to infinity. I suspect it does, and must, but can’t prove it:

After sieving to the n-th prime, Pn, the next candidate twin higher than Pn is always an actual twin. So, of the infinitude of remaining candidate twins after n rounds of sieving, the next candidate will not be removed until we reach, in successive rounds of sieving, the lower prime in the twin pair. Or in other words, there will be no lower candidate twin pairs that contain one or more composite numbers.

The other observation, which is I think an easy claim to defend, but I think has important implications in what I am trying to develop, is that in the (n+1)-th round of sieving, the only prime candidate between Pn^2 and P(n+1)^2 that can be removed as a candidate prime, not having been removed by prior rounds of sieving, is P(n+1)^2 itself. Therefore, at most one candidate twin pair can be removed between Pn^2 and P(n+1)^2, and at most two twin prime pairs less than or equal to P(n+1)^2 can be removed (i.e. those affected by P(n+1) and P(n+1)^2).

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## 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.