Can one show a beginning student how to use the $p$-adics to solve a problem?

Question

I recently had a discussion about how to teach $p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to $p$-adics because no one told them why the $p$-adics are useful.

As a graduate student in algebraic number theory, this question is easy to answer. But I'm wondering if there's a way to answer this question to someone who only knows very basic things about number theory and the $p$-adics. I'm thinking of someone who has learned over the course of a few days what the $p$-adic numbers are, what they look like, etc, but doesn't know much more.

I'm specifically wondering if there's any elementary problem that one can solve using $p$-adic numbers. It's okay if the answer is no, and that it takes time to truly motivate them (other than more abstract motivation).

Answer 1

While this may not be novel with respect to the $p$-adic numbers, one can show that $x^2-2=0$ has no solution in $\mathbb{Q}_5$, and therefore it follows that $\sqrt{2}$ is not rational. There are of course many other examples of this nature, perhaps one can find some more interesting ones.

Answer 2

An example for Hasse-Minkowski might be worth studying it, i.e., the binary quadratic form $5x^2 + 7y^2 − 13z^2$ has a non-trivial rational root since it has a $p$-adic one for every prime, and obviously also a real root.

Another example is the $3$-square theorem of Gauss: A positive integer $n$ is the sum of three squares if and only if $-n$ is not a square in $\mathbb{Q}_2$, the field of $2$-adic integers.

Answer 3

I'm not sure to what extent this addresses your question, but hopefully you find something interesting here. This actually was the way I first got introduced to the p-adics, so it was a good example for one person at least...

This is the last problem from the 2001 Bay Area Mathematical Olympiad:

For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1,2,\ldots,n\}$ such that $\tau(\tau(\tau(x)))=x$ for $x = 1,2,\ldots,n$. The first few values are $a_1 = 1,a_2 = 1,a_3 = 3,a_4 = 9$. Prove that $3^{334}$ divides $a_{2001}$.

This can be solved with elementary methods, but the result itself is not tight—in fact, $3^{445}$ divides $a_{2001}$. In general, $\nu_3 (a_n) \sim \frac{2}{9}n$.

As far as I know, this last fact requires working over the $3$-adics . The point is that $\sum_k a_k \frac{x^k}{k!}$, the exponential generating function for $a_n$, exactly equals $e^{x+\frac{1}{3}x^3}$, and finding an asymptotic expression $\nu_3 (a_n) \sim c n$ is equivalent to finding the radius of convergence, related by $R=3^{c-\frac{1}{2}}$ (here we use the fact that $\nu_3(k!)\sim \frac{1}{2}k$).

This is not necessarily the cleanest "application" of the p-adics, but I think it's an interesting place to start asking questions. For example, why does the power series for $e^x$ have 3-adic radius of convergence $3^{-\frac{1}{2}}$, but for $e^{x+\frac{1}{3}x^3}$ the radius increases to $3^{-\frac{1}{6}}$? For me, this yielded some intuition about the Artin-Hasse exponential $e^{x+\frac{1}{3}x^3+\frac{1}{3^2}x^{3^2}+\ldots}$, whose power series has radius of convergence 1.

Answer 4

My personal elementary favorites are:

• Prove that $$ \frac11+\frac12+\frac13+\cdots+\frac1n $$ is not an integer, if $n>1$.
• And the variant of proving that $$ \frac11+\frac13+\frac15+\cdots+\frac1{2n+1} $$ is not an integer, if $n\ge1$.

Both are resolved by using the non-archimedean $p$-adic triangle inequality for a suitable choice of $p$.