A relation of Kolomogrov complexity
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Given a positive strictly monotonically increasing infinite sequence $n_1, n_2, dots$ with Kolmogorov complexity
$$K(n_i)geqlceillog_2 n_irceil/2,.$$
If $q_i$ is the greatest prime number that divides $n_i$, I've been asked to show to
show that the set $Q={q_i}_{i>0}$ is infinite. How do I do this?
My idea was to assume if it is finite, then we can write all $n_i=p_1^{r_1}...p_k^{r_k}$. So all the the information is encoded by $(r_1,...,r_k)$. We can recover the number by a program merely with input $r_1,...,r_k$. And I want to deduce such program can have length shorter that the given function.
computer-science formal-languages
asked Nov 16 at 15:41
CO2
1429
add a comment |
up vote
0
down vote
favorite
Given a positive strictly monotonically increasing infinite sequence $n_1, n_2, dots$ with Kolmogorov complexity
$$K(n_i)geqlceillog_2 n_irceil/2,.$$
If $q_i$ is the greatest prime number that divides $n_i$, I've been asked to show to
show that the set $Q={q_i}_{i>0}$ is infinite. How do I do this?
My idea was to assume if it is finite, then we can write all $n_i=p_1^{r_1}...p_k^{r_k}$. So all the the information is encoded by $(r_1,...,r_k)$. We can recover the number by a program merely with input $r_1,...,r_k$. And I want to deduce such program can have length shorter that the given function.
computer-science formal-languages
asked Nov 16 at 15:41
CO2
1429
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given a positive strictly monotonically increasing infinite sequence $n_1, n_2, dots$ with Kolmogorov complexity
$$K(n_i)geqlceillog_2 n_irceil/2,.$$
If $q_i$ is the greatest prime number that divides $n_i$, I've been asked to show to
show that the set $Q={q_i}_{i>0}$ is infinite. How do I do this?
My idea was to assume if it is finite, then we can write all $n_i=p_1^{r_1}...p_k^{r_k}$. So all the the information is encoded by $(r_1,...,r_k)$. We can recover the number by a program merely with input $r_1,...,r_k$. And I want to deduce such program can have length shorter that the given function.
computer-science formal-languages
asked Nov 16 at 15:41
CO2
1429
Given a positive strictly monotonically increasing infinite sequence $n_1, n_2, dots$ with Kolmogorov complexity
$$K(n_i)geqlceillog_2 n_irceil/2,.$$
If $q_i$ is the greatest prime number that divides $n_i$, I've been asked to show to
show that the set $Q={q_i}_{i>0}$ is infinite. How do I do this?
My idea was to assume if it is finite, then we can write all $n_i=p_1^{r_1}...p_k^{r_k}$. So all the the information is encoded by $(r_1,...,r_k)$. We can recover the number by a program merely with input $r_1,...,r_k$. And I want to deduce such program can have length shorter that the given function.
computer-science formal-languages
computer-science formal-languages
asked Nov 16 at 15:41
CO2
1429
asked Nov 16 at 15:41
CO2
1429
asked Nov 16 at 15:41
CO2
1429
asked Nov 16 at 15:41
CO2
1429
asked Nov 16 at 15:41
CO2
1429
1429
add a comment |
add a comment |
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