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Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.

Mayank Pandey
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    How about using the Hasse-Minkowski theorem to show the existence of solutions for quadratic forms? Many elementary problems boil down to a quadratic equation, and one can use the Hasse-Minkowski theorem (p-adics) to show whether there is a solution. – Álvaro Lozano-Robledo Apr 16 '14 at 18:05
  • Can you give me an example? I am not very familiar with the Hasse-Minkowski theorem, because I don't really understand the wikipedia article. – Mayank Pandey Apr 17 '14 at 00:10
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    See http://math.stackexchange.com/questions/136206/show-that-sqrt1t-lies-in-mathbbz1-2-t/136288#136288 and http://mathoverflow.net/questions/81342/elementary-results-with-p-adic-numbers – KCd Apr 27 '14 at 02:25
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    See also http://math.stackexchange.com/questions/696669/can-one-show-a-beginning-student-how-to-use-the-p-adics-to-solve-a-problem/798540#798540 – KCd May 20 '14 at 21:39

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One can use $p$-adic integers for the question which positive integers are sum of three squares. For example, $n=7$ is not a sum of three squares, but $49=2^2+3^2+6^2$ is. The statement is that $n$ is a sum of three squares if and only if $-n$ is a square in $\mathbb{Q}_2$, the field of $2$-adic integers. This is the case if and only if $n$ is not of the form $4^{\ell}(8k+7)$ for non-negative integers $k,\ell$. (There is the Hasse-Minkowski theorem in the background, too). An example for Hasse-Minkowski is the following. $5x^2+7y^2-13z^2=0$ has a non-trivial real solution and a $p$-adic one for each prime $p$. Hence Hasse-Minkowski gives also a non-trivial rational solution, e.g., take $(x,y,z)=(3,1,2)$.

Dietrich Burde
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