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Let $n \gt 1$ and $$\left\lfloor\frac n 1\right\rfloor + \left\lfloor\frac n2\right\rfloor + \ldots + \left\lfloor\frac n n\right\rfloor = \left\lfloor\frac{n-1}{1}\right\rfloor + \left\lfloor\frac{n-1}{2}\right\rfloor + \ldots + \left\lfloor\frac{n-1}{n-1}\right\rfloor + 2$$ and $\lfloor \cdot \rfloor$ is the floor function. How to prove that $n$ is a prime?

Thanks in advance.

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3  
What does $\left \[\dfrac{a}n\right \]$ mean? –  user17762 May 10 '13 at 19:38
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Is this possible? Looks like the LHS is always 1 and the RHS is always greater than 2. –  Narut Sereewattanawoot May 10 '13 at 19:42
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@NarutSereewattanawoot: So since the assumption is always false, the conclusion follows whenever the assumption holds. ☺ –  Harald Hanche-Olsen May 10 '13 at 19:52
    
@N.S. It was 1 before the OP edited it. –  Narut Sereewattanawoot May 10 '13 at 19:59

2 Answers 2

up vote 12 down vote accepted

You know that

$$\left( \left\lfloor\frac n 1\right\rfloor - \left\lfloor\frac{n-1}{1}\right\rfloor \right)+\left( \left\lfloor\frac n2\right\rfloor - \left\lfloor\frac{n-1}{2}\right\rfloor\right) + \ldots + \left( \left\lfloor\frac{n}{n-1}\right\rfloor - \left\lfloor\frac{n-1}{n-1}\right\rfloor\right) + \left\lfloor\frac n n\right\rfloor=+2 \,.$$

You know that

$$\left( \left\lfloor\frac n 1\right\rfloor - \left\lfloor\frac{n-1}{1}\right\rfloor \right)=1$$ $$\left\lfloor\frac n n\right\rfloor =1$$ $$\left( \left\lfloor\frac n k\right\rfloor - \left\lfloor\frac{n-1}{k}\right\rfloor\right) \geq 0, \qquad \forall 2 \leq k \leq n-1 \,.$$

Since they add to 2, the last ones must be equal, thus for all $2 \leq k \leq n-1$ we have

$$ \left\lfloor\frac n k\right\rfloor - \left\lfloor\frac{n-1}{k}\right\rfloor = 0 \Rightarrow \left\lfloor\frac n k\right\rfloor = \left\lfloor\frac{n-1}{k}\right\rfloor $$

It is easy to prove that this means that $k \nmid n$. Since this is true for all $2 \leq k \leq n-1$, you are done.

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Empty of any defects. +1 –  Babak S. May 10 '13 at 20:04
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I think that in general we have $ \left\lfloor\frac n 1\right\rfloor + \left\lfloor\frac n2\right\rfloor + \ldots + \left\lfloor\frac n n\right\rfloor = \left\lfloor\frac{n-1}{1}\right\rfloor + \left\lfloor\frac{n-1}{2}\right\rfloor + \ldots + \left\lfloor\frac{n-1}{n-1}\right\rfloor + d$ where $d$ is the number of divisors of $n$...This seems to be provable exactly the same way. –  N. S. May 10 '13 at 20:48

i have a question if i define the function $ \sigma (x)= \sum_{n=1}^{x}d $ and 'd' is the function divisor isn't the problem $ \sigma(x)=\sigma(x-1)+2 $

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You are right this is a nice solution. $\lfloor \frac{n}{k} \rfloor$ counts how many numbers between $1$ and $n$ are multiple of $k$. Reordering you get exactly $\sum_{i=1}^n d(i)$. –  N. S. May 10 '13 at 20:53
    
To make it a formal proof: For each of the numbers between $1$ and $n$ list all the divisors. Then there are exactly $\sum_{i=1}^n d(i)$ numbers in the list.... Now, each $k$ appears exactly $\lfloor \frac{n}{k} \rfloor$ times in the list. Thus, there are exactly $\sum_{k=1}^n \lfloor \frac{n}{k} \rfloor$ numbers in the list. Thus $\sum_{k=1}^n \lfloor \frac{n}{k} \rfloor=\sum_{i=1}^n d(i)$. –  N. S. May 11 '13 at 15:26

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