Determination of position of non-primes and candidate primes – Continued


Now let’s illustrate the process I just described but using the primorial-based place value numbering system.

Consider the set of non-primes and possible primes within P3# = 5# = 30:

The “possible primes” are: 1, 7, 11, 13, 17, 19, 23, and 29, or:

0,0,1
1,0,1
1,2,1
2,0,1
2,2,1
3,0,1
3,2,1
4,2,1

Recall the meaning of “possible primes” is that these positions within all successive repetitions of groups of 30 natural numbers *may* have prime numbers at those positions, but those positions do not *have to* be prime. All other positions with repetitions of 30 can NEVER be prime, since they are all multiples of 2, 3 or 5. The primes we have already used to sieve, 2, 3 and 5, do not appear in this list but are known by now to be prime.

To obtain the set of possible primes within the next primorial, P4#, 210, we will repeat this pattern 6 more times, or in other words add 30 to each of these numbers 6 times, to yield 7 sets of 30 within 210, giving us the following set of numbers, A7:

0,0,0,1; 1,0,0,1; 2,0,0,1; 3,0,0,1; 4,0,0,1; 5,0,0,1; 6,0,0,1
0,1,0,1; 1,1,0,1; 2,1,0,1; 3,1,0,1; 4,1,0,1; 5,1,0,1; 6,1,0,1
0,1,2,1; 1,1,2,1; 2,1,2,1; 3,1,2,1; 4,1,2,1; 5,1,2,1; 6,1,2,1
0,2,0,1; 1,2,0,1; 2,2,0,1; 3,2,0,1; 4,2,0,1; 5,2,0,1; 6,2,0,1
0,2,2,1; 1,2,2,1; 2,2,2,1; 3,2,2,1; 4,2,2,1; 5,2,2,1; 6,2,2,1
0,3,0,1; 1,3,0,1; 2,3,0,1; 3,3,0,1; 4,3,0,1; 5,3,0,1; 6,3,0,1
0,3,2,1; 1,3,2,1; 2,3,2,1; 3,3,2,1; 4,3,2,1; 5,3,2,1; 6,3,2,1
0,4,2,1; 1,4,2,1; 2,4,2,1; 3,4,2,1; 4,4,2,1; 5,4,2,1; 6,4,2,1

From this set we will now remove multiples of 7 as follows: Take the set of possible primes within 30, namely 1, 7, 11, 13, 17, 19, 23, and 29, and multiply each by 7, giving us 7, 49, 77, 91, 119, 133, 161, 203, or the following set, B7:

0,1,0,1
1,3,0,1
2,2,2,1
3,0,0,1
3,4,2,1
4,2,0,1
5,1,2,1
6,3,2,1

Now remove the set B7 from A7 to get the set of possible primes within 7#, which we call PP7:

0,0,0,1; 1,0,0,1; 2,0,0,1; ——-; 4,0,0,1; 5,0,0,1; 6,0,0,1
——-; 1,1,0,1; 2,1,0,1; 3,1,0,1; 4,1,0,1; 5,1,0,1; 6,1,0,1
0,1,2,1; 1,1,2,1; 2,1,2,1; 3,1,2,1; 4,1,2,1; ——-; 6,1,2,1
0,2,0,1; 1,2,0,1; 2,2,0,1; 3,2,0,1; 4,2,0,1; 5,2,0,1; 6,2,0,1
0,2,2,1; 1,2,2,1; ——-; 3,2,2,1; 4,2,2,1; 5,2,2,1; 6,2,2,1
0,3,0,1; ——-; 2,3,0,1; 3,3,0,1; 4,3,0,1; 5,3,0,1; 6,3,0,1
0,3,2,1; 1,3,2,1; 2,3,2,1; 3,3,2,1; 4,3,2,1; 5,3,2,1; ——-
0,4,2,1; 1,4,2,1; 2,4,2,1; ——-; 4,4,2,1; 5,4,2,1; 6,4,2,1

Now wasn’t that easy? 🙂

Now if we want to know all primes less than 7^2 = 49, we can examine this list. All known primes 2, 3, 5 and 7 are prime, followed by the remaining “possible primes” less than 49, or 11, 13, 17, 19, 23, 29, 31, 37, 41, and 43, or:

0,1,2,1
0,2,0,1
0,2,2,1
0,3,0,1
0,3,2,1
0,4,2,1
1,0,0,1
1,1,0,1
1,1,2,1
1,2,0,1

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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