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The Prime Number Theorem gives a handy dandy little formula for estimating the density of primes below a given positive integer. (Overall, the ratio of primes relative to all of the other positive integers is vanishingly small.)
My question: Has anyone ever generalized the PNT to POWERS of primes? IOW, is there a handy dandy formula for estimating the density of all of the positive integral powers of primes that are less than a given positive integer?
My question: Has anyone ever generalized the PNT to POWERS of primes? IOW, is there a handy dandy formula for estimating the density of all of the positive integral powers of primes that are less than a given positive integer?
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Re: Powers of primes?
Sun, July 26, 2009 - 1:05 PMThe answer is rather easy:
If one wants to know the number of primes raised to the x power below y, an integer, then use the prime number theorem using the Floor(y^(1/x)) as the maximum integer.
Or did I misunderstand your question? -
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Re: Powers of primes?
Sun, August 2, 2009 - 12:07 AMI've had a mental block for the last several days. Today, I'm finally able to wrap my head around what you said. Thank you.
I was interested in the density of all positive integer powers of all primes--not just a single prime--below a given positive integer. A computationally intensive variation on the theme that you mentioned should work. Apparently there isn't a succinct formula yet.
A second question. I was checking out the taxonomy of integers at wikipedia, and did not see a category for positive integer powers of primes (including the primes themselves). So we have 2, 4, 8 etc; 3, 9, 27, etc; 5, 25, 125, etc; etc. Does this subset of the integers have a name? If not, how about ladder numbers? -
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Re: Powers of primes?
Sun, August 2, 2009 - 12:52 PM"A second question. I was checking out the taxonomy of integers at wikipedia, and did not see a category for positive integer powers of primes (including the primes themselves). So we have 2, 4, 8 etc; 3, 9, 27, etc; 5, 25, 125, etc; etc. Does this subset of the integers have a name? If not, how about ladder numbers? "
Not that I know of, the issue of density of powers of primes isn't of much general use (as far as I can tell) so no one has spent much time looking at them. The great thing about mathematics, is that amateurs have as much access to the problems as the pro's, play with this to your heart's content, maybe you'll find something that others are interested in.
BTW naming things is normally reserved for things that come-up repeatedly in the general community. For your own use, ladder numbers sounds like a nice short hand.
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