I am reading an article Modular Arithmetic by Richard Taylor. I have 2 questions:
For which $n$, $x^2+y^2=nz^2$ has nontravial solutions? What are the solutions?
A beautiful theorem of Hermann Minkowski and Helmut Hasse says that if $Q (X_1 , ..., X_d ) $ is any homogeneous quadratic polynomial in any number of variables with whole number coefficients, then
$$Q (X_1 , ..., X_d ) $$
has a non-zero solution in whole numbers if and only if it has a non-zero solution in all (real) numbers and a primitive solution modulo $m$ for all positive whole numbers $m$
How to prove the beautiful theorem? Could you recommend some books?
Many thanks in advance!