Density of the primes and the error introduced by the actual primes


There have been a few objections to my claims about converging product series representing, as n approaches infinity, the density within the set of natural numbers of primes or prime k-tuples. The objection is that with successive steps of sieving, a new “actual prime” is added to the set of primes, so the actual number of candidate primes, Kn, should be the value I’ve calculated for candidate k-tuples as Kn plus the number of actual primes, n.

As n approaches infinity, this error term becomes arbitrarily large, becoming infinity. However, this infinity is much smaller than the value of Kn, and vanishes in comparison when calculating the density. The following equations illustrate the concept:

Where i is an arbitrary number of primorials, Pn#, extending outward to infinity within the set of natural numbers used to calculate the density within the complete set of natural numbers.

In other words, the error term n, which diminishes in comparison with Pn# as n grows, vanishes completely in the calculation of density since it only affects the first primorial in an infinite series of primorials.

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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14 Responses to Density of the primes and the error introduced by the actual primes

  1. Hi Pete,

    Hmm, possibly, but these are the problems that spring to mind:

    “As n approaches infinity, this error term becomes arbitrarily large, becoming infinity. However, this infinity is much smaller than the value of Kn, and vanishes in comparison when calculating the density. The following equations illustrate the concept.”

    I don’t think so, because at the point where the error term reaches infinity, Kn reaches zero. It is hard to conceive this because of course at any point short of infinity, we can imagine an infinite string of k-tuple candidates. But we also know they are all eliminated at infinity.

    “In other words, the error term n, which diminishes in comparison with Pn# as n grows, vanishes completely in the calculation of density since it only affects the first primorial in an infinite series of primorials.”

    But the real list of primes is not a set of infinitely repeating primorials. It is an assymmetric shape with fractal tendencies, which can only really be seen as the first part of a single infinite “primorial” pattern. This is where the primes (and k-tuples) actually reside, not in any subsequent “repeats” of the pattern.

    I think that your equations probably do give a very rough estimate of the density of primes and k-tuples at certain levels, but with the proviso that an error term would be needed and that the estimate is rough.

  2. Sorry, yes, but I meant the density, which does tend to zero.

    And I think you need to be careful about this step…

    “As n approaches infinity, this error term becomes arbitrarily large, becoming infinity. However, this infinity is much smaller than the value of Kn”

    … because it is not comparing like for like. We know that the number of primes is higher than the number of k-tuples across particular large region. The only reason you can make this claim is because you are comparing different regions, the prime numbers from 0 to n with the much larger n#-sized primorial.

    Just to bring this out, the same logic would suggest that E (the set of even numbers) is growing much faster than O (the set of odd numbers), if we counted the odd numbers up to n and compared it to the even numbers in a n# primorial.

    • petequinn says:

      Thanks Hugh.

      There are, i think, A wide variety of logical inconsistencies that arise when we work with infinite sets. I’ll admit I’m no expert in that area (not by far) but I feel pretty confident I’ve got this aspect of the logic addressed correctly.

      When you say:

      “Just to bring this out, the same logic would suggest that E (the set of even numbers) is growing much faster than O (the set of odd numbers), if we counted the odd numbers up to n and compared it to the even numbers in a n# primorial.”

      I’m not sure what I’ve offered is really the same. Rather, it is akin to saying that the set of E is infinite because O is also infinite.

      There are a number of traps involved in reasoning with infinite sets. One could easily claim that the set of prime numbers is infinitely larger than the set of prime numbers, since for every prime, there is a corresponding primorial, and since the gap between successive primorials includes an ever increasing number of primes. Therefore for each prime, there are infinite other primes uniquely associated with that prime, hence the set of primes is infinitely larger than (indeed infinitely times as large as) the set of primes. Which seems obviously logically inconsistent. Or is it?

      Anyway I digress. I understand your objection, and perhaps you are correct, but I don’t think so.

      Cheers,

      Pete

  3. Hi Pete,
    Just on the point about O and E, the point is about comparing the size of different infinite sets. It’s not a perfect analogy,I know, but it’s an example of the dangers. Instead of that one, I’ll adapt another argument someone made to me once on this point.

    Imagine a sieve that casts out potential K-numbers. For each integer n, we consider the set of numbers up to n# and cast out n+1 as a potential K-number. The number of potential K-numbers within n# keeps growing, and we also know that no matter how high we go, the next number n will leave us with an infinity of potential K-numbers. If we ignore the error message given us by the density of actual special numbers in the set 1 …. n, then we might conclude that there is an infinite number of K-numbers.

  4. petequinn says:

    I’m not sure if I follow your logic. Is n# meant to be n-factorial? In any event, if you start sieving with 1, you eliminate all candidates immediately. I think I need to to explain this process a bit more clearly for me to be able to follow your logic and comment one way or another.

  5. Hi Pete,

    Yes, sorry, I should have said n!, not n#, typing too fast. The point is that you consider a larger set each time, just as we do with primorial patterns.

    And no, you don’t eliminate all candidates immediately. n only ever eliminates n+1. You only eliminate the next n+1 when you introduce the next n to the sieve. In this case it is intuitively obvious that all candidates will be eliminated eventually, but that is why it is obvious that there is a logical problem with this statement: “If you ignore the set 1 …. n, then we have overpowering evidence that there is an infinite number of K-numbers.”

    • petequinn says:

      OK thanks, I think I follow the recursive system now.

      You say: “”In this case it is intuitively obvious that all candidates will be eliminated eventually”

      I think that is not actually obvious, and would rather draw the opposite conclusion. Again, I think a subtle but important challenge when working with infinite sets is that we can’s use the same logic as applies with finite sets.

      And of course, it remains possible that my interpretation here is wrong. I think this at least clarifies the basis for our differences of opinion on the logic I’ve presented.

  6. I mean intuitively obvious as in “it’s clear that there are no K-numbers since, if we dispense with the sieve, we can see that for any candidate x, the number x-1 eliminates it.” That seems about as valid as Euclid’s proof of the infinitude of primes to me.

    In the case of twin primes we don’t have a clear, intuitive proof either way. From observation we strongly suspect it is not possible for the set of primes to eliminate all future twin prime candidates after a certain point. But to prove this conjecture, we need to prove that the set of twin prime candidates doesn’t at some point become like K-numbers in which all future candidates will be eliminated by future instances of ‘n’.

    “Again, I think a subtle but important challenge when working with infinite sets is that we can’t use the same logic as applies with finite sets.”

    I totally agree. That’s precisely why you can’t get to a proof of the twin prime conjecture from a comparison of the set of twin prime candidates with the set of prime candidates, since both tend to zero at infinity, and we can’t make safe deductions about about the infinite sets (primes/twin primes) from the logic of the finite sets (prime candidates/twin prime candidates *based on a finite list of primes*). I’m personally convinced there is an infinite number of twin primes, but it remains a very difficult thing to prove.

    • petequinn says:

      I mean intuitively obvious as in “it’s clear that there are no K-numbers since, if we dispense with the sieve, we can see that for any candidate x, the number x-1 eliminates it.”

      But you’ve created an additional arbitrary constraint that for every value of n, we have some higher value n!, so that for as long as we chase N+1 to infinity, the size of the set proceeds to grow exponentially, which places your problem in a completely different context I think.

  7. It’s exactly the same kind of constraint as looking at n#-length primorial patterns of k-tuple candidates and drawing conclusions about actual k-tuples. The point of it was to bring out the fact that you can only regard the ratio between n and Kn as trivial if you compare a set to one that grows much faster (as you do when you say “this infinity is much smaller than the value of Kn, and vanishes in comparison when calculating the density”.)

    But you can ignore that aspect of it and make the example simpler:

    Imagine a sieve that casts out potential K-numbers. For each postive integer n, n casts out n+100,000 as a potential K-number. As we work our way up through a sieve of integers, the number of K-number candidates is always infinite. The density of K-number candidates within the remaining set of integers is never reduced. And up to 100,000 we keep finding actual K-numbers.

    Remember, the point is not that this in any way shows that there isn’t an infinite number of twin primes. It’s just a demonstration that it is unsound to make logical inferences from the set of “candidates” to the set of actual numbers that the sieving process will “discover”.

    • petequinn says:

      Thanks for keeping the pressure on me Hugh, i appreciate being forced to think this through more rigourously. I’m not sure I accept your objection at face value, but I’m mulling it over, letting it roll around in my head. Meanwhile, I think I have a simpler way of laying this out, which i will attempt to elaborate shortly.

  8. Great, thanks for not taking offence at me hassling you.

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