Elimination of primes within primorials through successive sieving

Consider the following:

These tables show the progression of possible primes removed within the first three primorials, 2, 6 and 30, after sieving by 2, 3 and 5. Numbers are presented in the primorial notation, which I explained here:


Red numbers are those removed in the current round of sieving, orange were previously removed, and black remain as candidate primes.

For the second and third primorials I’ve presented the numbers in two different organizations – the first with numbers written from top to bottom, so that the lower primorial is presented in column format, and the second in traditional left to right enumeration, wrapping down to the next row again at the left.

We see different patterns emerge with the two different formats, which are more apparent when examining the next primorial, 210, shown both ways as follows:


In the first form, with lower primorials in column form, we have rows of non-primes and possible primes, with new non-primes (multiples of the prime currently being sieved) at seemingly random locations.

In the second form, all the new non-primes are in the left-most column (being multiples of 7), and this entire column is now ruled out as non-prime.

I will continue to mull these over to see if anything interesting occurs to me. Meanwhile, one might care to note that the emphasized box in the last table includes all numbers within 7# less than 7^2, so all possible primes in this box are prime.


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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