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Prove that\begin{equation} \pi(x)>\log_2\log_2x\end{equation}for all $x>1$.Proof:That is ,\begin{equation} 2^{\pi(x)}>\log_2x\end{equation}That is\b... 阅读全文
posted @ 2012-12-02 23:35
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Prove that\begin{equation} \pi(x)>\log_2\log_2x\end{equation}for all $x>1$.Proof:That is ,\begin{equation} 2^{\pi(x)}>\log_2x\end{equation}That is\b... 阅读全文
posted @ 2012-12-02 23:35
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Let $2=p_1<p_2<\cdots$ be the sequence of primes in increasingorder.Prove that\begin{equation} p_n\leq 2^{2^{n-1}}\end{equation}for all $n\geq 1$.Pro... 阅读全文
posted @ 2012-12-02 20:42
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Let $2=p_1<p_2<\cdots$ be the sequence of primes in increasingorder.Prove that\begin{equation} p_n\leq 2^{2^{n-1}}\end{equation}for all $n\geq 1$.Pro... 阅读全文
posted @ 2012-12-02 20:42
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Let $n_0\geq 6$,prove that if $\pi(n_0)\leq \frac{4n_0}{15}$,and $n=n_0+30k$,then $\pi(n)\leq \frac{4n}{15}$.证明:我寄希望于证明区间$(n_0,n_0+30k]$中素数个数不会多于$8k$.... 阅读全文
posted @ 2012-12-02 20:26
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Let $n_0\geq 6$,prove that if $\pi(n_0)\leq \frac{4n_0}{15}$,and $n=n_0+30k$,then $\pi(n)\leq \frac{4n}{15}$.证明:我寄希望于证明区间$(n_0,n_0+30k]$中素数个数不会多于$8k$.... 阅读全文
posted @ 2012-12-02 20:26
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Prove that $\pi(n)\leq \frac{n}{3}$ for $n\geq 33$.Proof:According to Eratosthenes's sieve method, when $n\geq33$,$\sqrt{33}\geq 5$.Then we delete al... 阅读全文
posted @ 2012-12-02 14:27
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Prove that $\pi(n)\leq \frac{n}{3}$ for $n\geq 33$.Proof:According to Eratosthenes's sieve method, when $n\geq33$,$\sqrt{33}\geq 5$.Then we delete al... 阅读全文
posted @ 2012-12-02 14:27
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Prove that $\pi(n)\leq \frac{n}{2}$ for $n\geq 8$.Proof:According to Eratosthenes's sieve method, when $n\geq8$,$\sqrt{n}\geq 2$.Then we delete all t... 阅读全文
posted @ 2012-12-02 13:28
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Prove that $\pi(n)\leq \frac{n}{2}$ for $n\geq 8$.Proof:According to Eratosthenes's sieve method, when $n\geq8$,$\sqrt{n}\geq 2$.Then we delete all t... 阅读全文
posted @ 2012-12-02 13:28
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Show that every prime number except 2 and 3 has a remainder of 1 or 5when divided by 6.Prove that there are infinitely many prime numberswhose remaind... 阅读全文
posted @ 2012-12-02 10:46
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Show that every prime number except 2 and 3 has a remainder of 1 or 5when divided by 6.Prove that there are infinitely many prime numberswhose remaind... 阅读全文
posted @ 2012-12-02 10:46
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There are infinitely many primes whose remainder is 3 when divided by 4.Proof:If there are only finite number of such primes,suppose they are\begin{eq... 阅读全文
posted @ 2012-12-02 10:26
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There are infinitely many primes whose remainder is 3 when divided by 4.Proof:If there are only finite number of such primes,suppose they are\begin{eq... 阅读全文
posted @ 2012-12-02 10:26
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