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What I am wanting to do, if it is possible is find unique matrix representations for the p-adic representation of numbers. So for example, say I have the number 1365 = 3*5*7*13. Now I could take various p-adic expansions of this, say $p=5$, I then get $1365 = 0*5^0+3*5^1+4*5^2+0*5^3+2*5^4$ which I then have the set {0,3,4,0,2} as the coefficients. I wonder if there is a way to write this, (in this case) as a 3x3 matrix, that being $$\mbox{} \left[\begin{array} \\ 0 & 3 & 4 \\ 3 & 4 & 0 \\ 4 & 0 & 2 \end{array} \right]$$ where the rows would be of the form $p^0,p^1,p^2$ as well as the columns,

$$\rho_{5} = \mbox{} \left[\begin{array} \\ x_{11}p^0 & x_{12}p^1 & x_{13}p^2 \\ x_{21}p^1 & x_{22}p^2 & x_{23}p^3 \\ x_{31}p^2 & x_{32}p^3 & x_{33}p^4 \end{array} \right]$$

note then that the element $x_{22}$ would be the co-efficient of 4 as multiplying the second element of the row and of the column we would get an element $p^1*p^1 = p^2$ which has a co-efficient of 4. Likewise $p^2*p^2 = p^4$ has the co-efficient 2. Would this matrix would be unique from any other number and choice of prime. Another example would be the same number $1365 = 16*19^0+14*19^1+3*19^2$ The smallest matrix (up to the needed $p^n = 19^2$ would be $$\rho_{19} = \mbox{} \left[\begin{array} \\ 16 & 14 \\ 14 & 3 \end{array} \right] = \mbox{} \left[\begin{array} \\ 16 & 14 & 3\\ 14 & 3 & 0 \\ 3 & 0 & 0 \end{array} \right]$$ in my trying to represent it in a 3x3 format so as to do multiplication.

If I multiply these "potential" representation matrices I get $$\mbox{} \left[\begin{array} \\ 0 & 3 & 4 \\ 3 & 4 & 0 \\ 4 & 0 & 2 \end{array} \right] \mbox{} \left[\begin{array} \\ 16 & 14 & 3\\ 14 & 3 & 0 \\ 3 & 0 & 0 \end{array} \right] = \mbox{} \left[\begin{array} \\ 54 & 9 & 0\\ 104 & 54 & 9 \\ 70 & 56 &12 \end{array} \right]$$ Which I suppose would have to somehow relate to $1365*1365$ It is not symmetric though.

Any idea on what I am looking for and/or how to make this happen? In the end I want to be able to represent numbers p-adically but in matrix format.

Thanks for your insight,

Brian

Relative0
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  • This appears to have some relation to base $p$, and no relation to $p$-adic numbers. – vadim123 Apr 30 '14 at 02:35
  • I suppose it does, I am trying to figure them (p-adic) numbers out and how to translate natural numbers to it - which is why I started with finding (or at least think I found) the p-adic expansion (according to wikipedia) $\sum_{i=0}^{\infty} a_i p^i$. I understand that numbers are related based on their p-adic distance vs. Euclidean distance, am not sure how to get there though in representing them in matrix form and if that representation would even be unique. – Relative0 Apr 30 '14 at 02:49
  • With these matrices you're obviously trying to arrange the base-p coefficients of naturals for some purpose. Is the arrangement you described meaningful in some way? What are you trying to accomplish with the arrangement? What's wrong with just writing the coefficients sequentially in a list, exactly as in a power series? Why bother writing the same list over and over again, only deleting the first number in each case? That seems redundant and pointless to me. – anon Apr 30 '14 at 23:42
  • It seems that I could do without the "p-adic" representation for my purposes, and things seems simpler with tensoring the various matrices. There is an issue still with the result but invite anyone to check out: http://math.stackexchange.com/questions/776506/representing-natural-numbers-as-matrices-by-use-of-otimes . I will delete this question - thanks for taking the time to try and understand what I am looking to do. – Relative0 May 01 '14 at 02:32

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