Some polynomials with inputs in the real numbers always take non-negative values; an easy example is x2 + y2. Hilbert’s 17th problem asks whether such a polynomial can always be written as the sum of squares of rational functions (a rational function is the quotient of two polynomials). In 1927, Emil Artin solved the question in the affirmative.
The Formula to Get 42 Billion Digits of π
While writing "7 Utterly Well-written Math Books About Pi," I found a very interesting math formula that will give you 42 consecutive digits of π accurately but is still wrong.