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Prove that if $p$ and $q$ are twin primes greater than 3,then $p+q$ isdivisible by 12.Proof:According to Elementary Methods in Number Theory Exercise ... 阅读全文
posted @ 2012-11-29 23:46
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Prove that if $p$ and $q$ are twin primes greater than 3,then $p+q$ isdivisible by 12.Proof:According to Elementary Methods in Number Theory Exercise ... 阅读全文
posted @ 2012-11-29 23:46
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The prime numbers $p$ and $q$ are called twin primes if$|p-q|=2$.Let $p$ and $q$ be primes.Prove that $pq+1$ is a square ifand only if $p$ and $q$ are... 阅读全文
posted @ 2012-11-29 19:45
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The prime numbers $p$ and $q$ are called twin primes if$|p-q|=2$.Let $p$ and $q$ be primes.Prove that $pq+1$ is a square ifand only if $p$ and $q$ are... 阅读全文
posted @ 2012-11-29 19:45
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Find all primes $p$ such that $29p+1$ is a square.Proof:\begin{equation} 29p+1=t^2\end{equation}So\begin{equation} 29p=(t+1)(t-1)\end{equation}Both ... 阅读全文
posted @ 2012-11-29 17:36
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Find all primes $p$ such that $29p+1$ is a square.Proof:\begin{equation} 29p+1=t^2\end{equation}So\begin{equation} 29p=(t+1)(t-1)\end{equation}Both ... 阅读全文
posted @ 2012-11-29 17:36
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Prove that $n^5-n$ is divisible by 30 for every integer $n$.Proof:\begin{equation} n^5-n=(n-1)n(n+1)(n^2+1)\end{equation}\begin{equation} 6|(n-1)n(n... 阅读全文
posted @ 2012-11-29 17:00
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Prove that $n^5-n$ is divisible by 30 for every integer $n$.Proof:\begin{equation} n^5-n=(n-1)n(n+1)(n^2+1)\end{equation}\begin{equation} 6|(n-1)n(n... 阅读全文
posted @ 2012-11-29 17:00
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Let $n\geq 2$,prove that $(n+1)!+k$ is composite for$k=2,\cdots,n+1$.This shows that there exists arbitrarily longintervals of composite numbers.Proof... 阅读全文
posted @ 2012-11-29 16:28
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Let $n\geq 2$,prove that $(n+1)!+k$ is composite for$k=2,\cdots,n+1$.This shows that there exists arbitrarily longintervals of composite numbers.Proof... 阅读全文
posted @ 2012-11-29 16:28
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Prove that $n,n+2,n+4$ are all primes if and only if $n=3$.Proof:$\Rightarrow$:$n$ must be an odd number,let $n=2k+1,k\in\mathbf{N^+}$.So $n,n+2,n+4$ ... 阅读全文
posted @ 2012-11-29 15:56
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Prove that $n,n+2,n+4$ are all primes if and only if $n=3$.Proof:$\Rightarrow$:$n$ must be an odd number,let $n=2k+1,k\in\mathbf{N^+}$.So $n,n+2,n+4$ ... 阅读全文
posted @ 2012-11-29 15:56
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Compute the standard factorization of $15!$.Solve:\begin{equation} \sum_{r=1}^{\infty}[\frac{15}{2^r}]=11\end{equation}\begin{equation} \sum_{r=1}^{... 阅读全文
posted @ 2012-11-29 02:43
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Compute the standard factorization of $15!$.Solve:\begin{equation} \sum_{r=1}^{\infty}[\frac{15}{2^r}]=11\end{equation}\begin{equation} \sum_{r=1}^{... 阅读全文
posted @ 2012-11-29 02:43
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Factor $10^k+1$ into a product of primes for $k=1,2,3,4,5$.Solve:When $k=1$,\begin{equation} 11=11\end{equation}.When $k=2$,\begin{equation} 101=101... 阅读全文
posted @ 2012-11-29 02:04
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Factor $10^k+1$ into a product of primes for $k=1,2,3,4,5$.Solve:When $k=1$,\begin{equation} 11=11\end{equation}.When $k=2$,\begin{equation} 101=101... 阅读全文
posted @ 2012-11-29 02:04
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Factor 51948 into a product of primes.Solve:$$ 51948=2^2\times 3^3\times 13\times 37 $$ 阅读全文
posted @ 2012-11-29 01:20
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Factor 51948 into a product of primes.Solve:$$ 51948=2^2\times 3^3\times 13\times 37 $$ 阅读全文
posted @ 2012-11-29 01:20
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