If I had explained it further, then the proof would have been longer! Right, we need the Fundamental Theorem of Arithmetic, that each integer factors uniquely

If I had explained it further, then the proof would have been longer! Right, we need the Fundamental Theorem of Arithmetic, that each integer factors uniquely into a product of prime numbers; right, that statement leaves out a little about the order of the factors. Well, 2 is prime number. So however many factors of 2 q has, q^2 has twice that many and, thus, an even number of factors of 2. Same argument for p^2; it also has an even number of factors of 2. But then 2 q^2 has an odd number of factors of 2. Uh, more generally, we can keep arithmetic simple or make it complicated. E.g., multiplication of integers is associative so that (2 x 3) x 4 = 2 x (3 x 4) The intermediate results are different, but the final results are the same. So there is a little something there (matrix multiplication is also associative, with more there -- right, easily function composition is associative so regard a matrix as a function and matrix multiplication as function composition). We can keep the Fundamental Theorem of Arithmetic simple, e.g., ask a 4rd grader, or make it complicated: To make arithmetic complicated, we can take a college course in abstract algebra that proves all this. Then for still more complicated we can go through axiomatic set theory before the abstract algebra course. Then in the set theory course we can get to undecidable issues. I went through all that. Instead, what I wanted was differential geometry as background for general relativity, partial differential equations for the various applications, but I'd put up with exterior algebra, of course the inverse and implicit function theorems (proved in Fleming's book and an exercise in Rudin's), fluid flow, Maxwell's equations, Schroedinger's equation, Fourier theory, Lebesgue integration, Leibniz's rule (differentiation under the integral sign; there's a simple proof if assume enough continuity and a more general proof from Lebesgue theory), how come, with Lagrange, action is stationary, on and on. I was willing to take arithmetic as okay!