# Math Help - 7-adic distance in Q7

1. ## 7-adic distance in Q7

Find the 7-adic distance between -1/3 and the square root of 305 in Q7

My solution is write the 7-adic expansion of -1/3 and square root of 305

the 7-adic expansion of -1/3 := 2 +2*7+2*(7^2)+.... =2.2222222(index 7)

the 7-adic expansion of square root of 305:= 2.2222222(index7) this is one representation of root of 305 but there is another representation of 7-adic expansion

Hence they have the same 7-adic expansion . therefore the distance is 0 in Q7?

Is my attemption right? ( i am not sure whether i caculated these two 7-adic expansion in a right way)

2. ## Re: 7-adic distance in Q7

$305_{10} = 614_7$

So, you are looking for $k = \sum_{n\ge 0}d_n 7^n$ such that $k^2 = 614_7$. So, suppose $d_0=2$ (it can be 2 or 5). Then $2d_0d_1 \equiv 1 \pmod{7}$, so $4d_1 \equiv 1 \pmod{7}$. That gives $d_1 = 2$, and a carry of 1. So, $1+2d_0d_2 + d_1^2 = 4d_2 + 5 \equiv 6 \pmod{7}$. This gives $d_2=2$ as you suggested. Again, the carry is 1. Then, $1+2d_0d_3 + 2d_1d_2 = 4d_3+9 \equiv 0 \pmod{7}$, so $d_3 = 3$ and the carry is 3. Next, $3+2d_0d_4 + 2d_1d_3 + d_2^2 = 3+4d_4 + 12 + 4 = 19+4d_4 \equiv 0 \pmod{7}$. So, $d_4 = 4$, and the carry is 5. So, I don't know how you calculated the square root of 305, but I got a very different result.

3. ## Re: 7-adic distance in Q7

yes ur right, i got some mistakes of my caculation. now, i got the new representation for root 305 should be right: 2.22345... or
5.4432....

and my question is how to caculate the distance between -1/3
and root of 305 in Q7 (i.e. w.r.t the norm|.|p) Hints for the question is:
compute first few terms of 7-adic expansion of -1/3 and root 305 .

Now , i dont know how to consider the distance..

4. ## Re: 7-adic distance in Q7

The easiest way to check if a solution is correct is to use evaluate its geometric sum: $\sum_{n \ge 0}2\cdot 7^n = 2\sum_{n\ge 0}7^n = 2\dfrac{1}{1-7} = 2\left(\dfrac{1}{-6}\right) = -\dfrac{1}{3}$. So, you found the correct representation for $-1/3$. So, $\left|-\dfrac{1}{3} - \sqrt{305} \right|_7$ is the minimum distance between the possible square roots of $305$. So, if $k = \ldots 43222._7$ and $-k = \ldots 34555._7$, and both of them squared give $614_7 = 305_{10}$, the 7-adic distance is $\min \left\{ \left|\ldots 22222._7 - \ldots 43222._7 \right|_7, \left| \ldots 22222._7 - \ldots 34555._7 \right|_7 \right\} = \min \left\{7^{-3},7^0 \right\}$

5. ## Re: 7-adic distance in Q7

sorry, do you mean if i continue to do the method for more digits of 2.22345... or 5.4432.... then i can get k and -k as ur post?? but how could u ensure that for 2.22345... , u can only get 5 on the digits after 4？？？

6. ## Re: 7-adic distance in Q7

You are writing $2.22345...$. That is not a notation I am familiar with. I am used to $\ldots 43222._7$ which means $2 + 2\cdot 7 + 2\cdot 7^2 + 3\cdot 7^3 + 4\cdot 7^4 + \cdots$. I suspect they mean the same thing. I am using only digits to the left of the decimal place. You are using digits to the right of the decimal place. I learned that $2.2_7 = 2\cdot 7^{-1} + 2\cdot 7^0$. Anyway, using your notation, you want $\min\left( \left|2.22222 - 2.22345\right|_7, \left|2.22222 - 5.44321 \right|_7\right)$. Note that $5.44321 = -2.22345$.

7. ## Re: 7-adic distance in Q7

Thats clear for me now. thanks a lot

8. ## Re: 7-adic distance in Q7

How did you calculate the 7-adic expansion of square root 305? i honestly cannot do it

9. ## Re: 7-adic distance in Q7

Originally Posted by lahuxixi
How did you calculate the 7-adic expansion of square root 305? i honestly cannot do it
me too

10. ## Re: 7-adic distance in Q7

Calculate it digit-by-digit. I went through it in post #2.

11. ## Re: 7-adic distance in Q7

I don't need the answer to the question of this thread. I want to ask you about the method of calculating the expansion for other cases. Thank you

12. ## Re: 7-adic distance in Q7

Originally Posted by SlipEternal
$305_{10} = 614_7$
what does this mean?