• Mathematicians Use Numbers Differently From The Rest of Us, by Muller
• p-adics, by Eric Rowland

Build up p-adics in small steps. Maybe background steps:

• Pythagorean Triples – Charlie knows how to generate them
• number systems w/different bases – many know base 2, 8, 16; Muller uses base 3 (3-adics) and Rowland uses base 5 (5-adics)
• harmonic series, summation, simple to prove; Muller uses the summation. And he uses the series as a familiar illustration of convergence to the right to contrast the possibility of 10-adic convergence to the left. And the same for 3-adic numbers.
• Diophantine equations
• Modulo arithmetic

There is mention of modular arithmetic. And mention that it is useful when testing Fermat's theorem for prime numbers (not his Last Theorem). Fermat's theorem says for any prime p if you take any relatively prime number n to the power p-1 and divide by p you will always get a remainder of 1. Modulo arithmetic makes it possible to do in your head the calculation for three-digit prime numbers. For example, I can do it for p = 101 and n = 2 . Harder to do for n=3 or 4 or 5 …).

Modular arithmetic is also useful for solving some Diophantine equations. For example, find integers that solve 71x+37y = 3000. (I can do this without modular arithmetic.)

And eventually we get to the point where we show how Muller solved x^2 + x^4 +x^8 = y^2 using 3-adics.

Mark Rober's Commencement Speech (20m)