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mfb

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What do you mean by "generated"?'ve noticed that for every prime and every composite number on the number line that is not a perfect power, ( I am referring to positive whole numbers only), that there are infinitely many perfect powers (PPs) that can be generated using that number.

Both the number of primes and the number of perfect powers is infinite - their number is the same. What you are looking at is the relative frequency within the first N numbers, where there are more primes than perfect powers. Deriving very large perfect powers from small primes doesn't do anything to help there.

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Stephen Tashi

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Why does it suggest that? You are considering finite intervals. For example, if we generate the powers of 5, they are more and more widely spaced apart. If the powers of numbers were equally spaced on the number line ( as are the multiples of numbers) then intuition would say that they would have a great "density" in a given finite interval. But since the perfect powers don't have constant "density", over intervals, I don't see that intuition can decide about who wins as far as the density perfect powers versus non-perfect powers in a finite interval.This suggests that the PPs should swamp the primes plus the non-powered composites.

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'Generated' what I was trying to suggest is that one could take any number, 3 for example, and square it, cube it ....... infinitely many times, creating an infinite group of perfect numbers to match only one prime. the number of powers of three greatly out number one which is the quantity of the number three. Then repeat for each single prime or other composite which has not yet been raised to a power.

The whole point I was getting at was that the powers spread out too fast creating the impression, to me , that there are fewer powers then other numbers. rather than the other way around which I had expected. Thanks to all.You have given me lots to digest.

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Stephen Tashi

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fresh_42

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It seems you all know. What is a perfect power?

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Stephen Tashi

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I'm assuming its a natural number that is equal to a natural number power of some other (different) natural number.It seems you all know. What is a perfect power?

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fresh_42

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So ##P(10)=3##, i.e. ##a^n## is one and ##a^nb^m## is not? Isn't ##P(100)=12## a bit low then?I'm assuming its a natural number that is equal to a natural number power of some other (different) natural number.

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Stephen Tashi

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Yes, I want ##P(10) = 3##, the cardinality of ##\{2^2,2^3, 3^2\}##. So, yes, ##a^nb^m## need not be a "perfect power". We'll see what the others say.So ##P(10)=3##, i.e. ##a^n## is one and ##a^nb^m## is not?

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Stephen Tashi

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It would be nice to have a recursion: given ##P(N)## find ##P(N+1)##. However, such a recursion might have a lot of if...then cases. Is there a theory that treats recursions like that?

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mfb

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Most perfect powers below N are squares, and the fraction of other perfect powers goes to zero for N to infinity. There are N^(1/2) squares, N^(1/3) cubes and so on, woth some overlap. Up to 1 million there are 1000 squares, 100 cubes, and just 10 numbers in both categories. There are also a few higher powers, e.g. 15 fifth powers, 3 of them (1^10, 2^10, 3^10) overlapping with the squares, and two overlapping with the cubes (1^15, 2^15). With inclusion-exclusion you can count them in a systematic way.

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Yes, this is a paradox of infinite sets. A simpler example is to consider the set of natural numbers and the set of even numbers:

'Generated' what I was trying to suggest is that one could take any number, 3 for example, and square it, cube it ....... infinitely many times, creating an infinite group of perfect numbers to match only one prime. the number of powers of three greatly out number one which is the quantity of the number three. Then repeat for each single prime or other composite which has not yet been raised to a power.

The whole point I was getting at was that the powers spread out too fast creating the impression, to me , that there are fewer powers then other numbers. rather than the other way around which I had expected. Thanks to all.You have given me lots to digest.

##\mathbb{N} = \{1, 2, 3, \dots \}##

##S = \{2, 4, 6, \dots \}##

Clearly, we can put ##\mathbb{N}## into one-to-one correspondence with the set ##S##, in that sense, both ##\mathbb{N}## and ##S## have the same "number" of members and are sets of the same "size". More properly, they are sets of the same

Equally, we can see that ##S## is a proper subset of ##\mathbb{N}##. In that sense ##\mathbb{N}## contains "twice as many" members as ##S##.

Your idea takes this a step further. If we take a prime ##p## and define the set:

##P_p = \{p, p^2, p^3 \dots \}##

Then we cleary have a set that can be put into one-to-one correspondence with ##\mathbb{N}##. Just associate ##p^n## with the number ##n##.

If we take the union of all these sets, we have:

##P = P_2 \bigcup P_3 \bigcup P_5 \dots ##

And, yet, ##P \subset \mathbb{N}##

What we've done is shown that ##\mathbb{N}## can contain a whole infinite sequence of sets, all the same cadinality as itself, and still have plenty left over: "most" numbers, after all, are not prime powers.

To put it another way, ##P## is the union of an infinite collection of sets, all the same cardinality as ##\mathbb{N}##, but still only has the same cardinality as ##\mathbb{N}##.

These sort of paradoxes were sorted out by a mathematician called Georg Cantor. If you want to look him up and read more about this.

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jbriggs444

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You might also read about asymptotic density which is roughly what @Stephen Tashi was getting at in #12 above.

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Will do. Thx.You might also read about asymptotic density which is roughly what @Stephen Tashi was getting at in #12 above.

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