p-adic Numbers
We close with one of the strangest and most powerful ideas in number theory. The p-adic
numbers grow out of a wholly different sense of when two numbers are "close" — and that one
change of perspective gives the primes a geometry of their own.
A new notion of size
Fix a prime p. The p-adic distance declares a number
small when it is divisible by a high power of p. So
p, p^2, p^3, \dots march toward zero:
|p^{k}|_{p} = p^{-k} \longrightarrow 0 \quad\text{as } k \to \infty.
Modulo 5, the numbers 25 and
50 are "tiny" while 3 is "large". This is a
genuine distance — it satisfies an even stronger triangle inequality — and completing the rationals
with respect to it gives the field \mathbb{Q}_p of p-adic numbers, exactly
as completing them the usual way gives \mathbb{R}.
Numbers as infinite expansions
Just as reals have decimal expansions going infinitely to the right, p-adic numbers have
expansions in powers of p going infinitely to the left:
x = a_0 + a_1 p + a_2 p^2 + a_3 p^3 + \cdots, \qquad 0 \le a_i < p.
These series converge p-adically because the terms shrink. The arithmetic stitches together
all the congruences
modulo p, p^2, p^3, \dots at once — a p-adic integer is precisely a
coherent choice of residue to every power of p.
The local–global principle
Why build this? Because it makes "solve modulo every power of p" into a
single statement: a solution exists p-adically. Hasse's local–global principle then
says a problem (for certain equations) has a solution in the rationals iff it has one in
\mathbb{R} and in every \mathbb{Q}_p — solve it
everywhere "locally" and you've solved it "globally".
With that, number theory comes full circle: the
divisibility
and congruences
we started with reappear as the very geometry of the p-adic world — a fitting end to the
queen of mathematics.