Probability that a natural number is a k-th power
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I would like to determine the probability that a natural number is a k-th power. It is quite straightforward to see that the probability for a natural number less than N to be a I-that power is
$$N^{1/k-1}$$
However I do not understand what to do to extend it for N going to infinity. I essentially though of cutting dyadically or so, and sum the « local » probability above...
probability number-theory elementary-number-theory
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add a comment |
$begingroup$
I would like to determine the probability that a natural number is a k-th power. It is quite straightforward to see that the probability for a natural number less than N to be a I-that power is
$$N^{1/k-1}$$
However I do not understand what to do to extend it for N going to infinity. I essentially though of cutting dyadically or so, and sum the « local » probability above...
probability number-theory elementary-number-theory
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1
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There is no such thing as a uniformly random natural number.
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– Hagen von Eitzen
Dec 9 '18 at 5:14
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I think that by probability, you mean the limit of #${ n leq N | n text{ is a $k$-th power}} / N$ for $N to infty$? You should specify this in your post.
$endgroup$
– Tki Deneb
Dec 9 '18 at 11:01
add a comment |
$begingroup$
I would like to determine the probability that a natural number is a k-th power. It is quite straightforward to see that the probability for a natural number less than N to be a I-that power is
$$N^{1/k-1}$$
However I do not understand what to do to extend it for N going to infinity. I essentially though of cutting dyadically or so, and sum the « local » probability above...
probability number-theory elementary-number-theory
$endgroup$
I would like to determine the probability that a natural number is a k-th power. It is quite straightforward to see that the probability for a natural number less than N to be a I-that power is
$$N^{1/k-1}$$
However I do not understand what to do to extend it for N going to infinity. I essentially though of cutting dyadically or so, and sum the « local » probability above...
probability number-theory elementary-number-theory
probability number-theory elementary-number-theory
asked Dec 9 '18 at 4:58
TheStudentTheStudent
3137
3137
1
$begingroup$
There is no such thing as a uniformly random natural number.
$endgroup$
– Hagen von Eitzen
Dec 9 '18 at 5:14
$begingroup$
I think that by probability, you mean the limit of #${ n leq N | n text{ is a $k$-th power}} / N$ for $N to infty$? You should specify this in your post.
$endgroup$
– Tki Deneb
Dec 9 '18 at 11:01
add a comment |
1
$begingroup$
There is no such thing as a uniformly random natural number.
$endgroup$
– Hagen von Eitzen
Dec 9 '18 at 5:14
$begingroup$
I think that by probability, you mean the limit of #${ n leq N | n text{ is a $k$-th power}} / N$ for $N to infty$? You should specify this in your post.
$endgroup$
– Tki Deneb
Dec 9 '18 at 11:01
1
1
$begingroup$
There is no such thing as a uniformly random natural number.
$endgroup$
– Hagen von Eitzen
Dec 9 '18 at 5:14
$begingroup$
There is no such thing as a uniformly random natural number.
$endgroup$
– Hagen von Eitzen
Dec 9 '18 at 5:14
$begingroup$
I think that by probability, you mean the limit of #${ n leq N | n text{ is a $k$-th power}} / N$ for $N to infty$? You should specify this in your post.
$endgroup$
– Tki Deneb
Dec 9 '18 at 11:01
$begingroup$
I think that by probability, you mean the limit of #${ n leq N | n text{ is a $k$-th power}} / N$ for $N to infty$? You should specify this in your post.
$endgroup$
– Tki Deneb
Dec 9 '18 at 11:01
add a comment |
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1
$begingroup$
There is no such thing as a uniformly random natural number.
$endgroup$
– Hagen von Eitzen
Dec 9 '18 at 5:14
$begingroup$
I think that by probability, you mean the limit of #${ n leq N | n text{ is a $k$-th power}} / N$ for $N to infty$? You should specify this in your post.
$endgroup$
– Tki Deneb
Dec 9 '18 at 11:01