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Mathematics

p-adic Numbers and Their Role in Solving Diophantine Equations

Quick fact

The p-adic numbers were introduced by Kurt Hensel in 1897, and they give a completely different notion of distance on rationals: two numbers are p-adically close if their difference is divisible by a large power of p. This leads to the surprising fact that a Diophantine equation may have no integer solution, yet have solutions modulo every prime power, illustrating the subtlety of the Hasse principle.

Why this is interesting

You know that numbers are close when their decimal expansions agree for many digits. What if we redefined 'closeness' so that numbers are close when they differ by a multiple of a high power of a prime? This simple twist creates a whole new number system that helps solve puzzles about integers.