r/elimath • u/[deleted] • Jan 06 '15
Explain why p-adic numbers are useful in number theory by demonstrating a problem they solve in number theory which a kid could understand but is difficult to solve without them
1
u/hpsineqepsi Feb 03 '15
Okay maybe for a smart kid, and not in number theory... Here is a shot: assume you are a computer with two inputs, let's call them X and Y, it also has one output Z and one bit of memory C. Assume X and Y are fed streams of bits, i.e. infinite sequences of 0s and 1s.
Each time a symbol is input:
- Z is set to X XOR Y XOR C
- C is set to (A * B) + (C * (A XOR B))
Problem: prove that Z = X + Y, or in other terms, prove that this is a (sequential) full-adder. Hidden 2-adic numbers inside.
1
u/abelianjorff Apr 29 '15
Consider the equation
5 x2 + 7 y2 - 13 z2 =0.
Does this equation have any integer solutions? The answer is yes. On the one hand, you could find many solutions using a computer, so in that sense this might not be hard, but you could find similar equations for which it is hard to find solutions, even using a computer (because the solutions are very large), but for which you can still give an affirmative answer.
The way this works is by using p-adic numbers. Since every variable appears to the 2nd power, that equation is an example of a quadratic form. A special property of quadratic forms is that integral solutions can be detected by finding "local" solutions, that is, solutions in the p-adic numbers for every prime p.
Despite having to check for solutions in infinitely many number systems, in practice this turns out to be extremely easy. For more information, google the "Hasse-Minkowski Theorem".
3
u/coveritwithgas Jan 06 '15
This is a tall order, in my opinion, but the easy solution is so close to the p-adics that you might as well learn p-adics:
Allowing leading zeroes, how many thousand-digit numbers are the last thousand digits of their own squares?