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Math Help - 7-adic distance in Q7

  1. #1
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    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)
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  2. #2
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    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.
    Last edited by SlipEternal; November 19th 2013 at 03:36 PM.
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  3. #3
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    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..
    Last edited by Sonifa; November 19th 2013 at 04:44 PM.
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  4. #4
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    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\}
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  5. #5
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    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???
    Last edited by Sonifa; November 19th 2013 at 05:10 PM.
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  6. #6
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    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.
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  7. #7
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    Re: 7-adic distance in Q7

    Thats clear for me now. thanks a lot
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  8. #8
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    Re: 7-adic distance in Q7

    How did you calculate the 7-adic expansion of square root 305? i honestly cannot do it
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    Re: 7-adic distance in Q7

    Quote Originally Posted by lahuxixi View Post
    How did you calculate the 7-adic expansion of square root 305? i honestly cannot do it
    me too
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  10. #10
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    Re: 7-adic distance in Q7

    Calculate it digit-by-digit. I went through it in post #2.
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    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
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  12. #12
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    Re: 7-adic distance in Q7

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