Proof isn't always possible

A black box and a function might have very little difference between them. What might make the term “black box” definitive? As formative thinking, a black box is characterized by accomplishing its operation without affecting ANYTHING outside its internal mechanisms; perhaps this includes the idea that it cannot be observed. A “function,” is defined as “a reusable block of code that a specific task. It accepts input, processes it and “returns” a result.” My first thought on this, is that functions can return intermediate values, and black boxes might be barred from iteration or recursion by definition.

I was privileged to show Newton’s Method in integer arithmetic. Newton wanted to use fractions to see more decimal places, and the duration between his efforts and my good fortune, attests to the difficulty of maintaining objectivity or gaining new perspective. Here I must document that my breakthrough hasn’t solved the problem completely! (Quelle surprise!) The method I develop requires raising very large numbers to appreciably large powers. The actual integers become impractical very quickly. Note: It is fairly practical to get up to six (6) iterations for small n on numerals 1-9, with Python 3. Use this to estimate efficiency- how many decimals are becoming significant? I stipulate further that it is impractical to guarantee an exact margin of error, using my solution. Having supplied an answer to some algebra problem requiring Newton’s Method for its solution, I want to close by suggesting this: If we still want to see more decimal places, is there material advantage to be fo...