About an exercise appearing in TAOCP volume one's "Notes on the Exercises".

Question

Hi,

There is a question in TAOCP vol 1, in "Notes on Exercises" section, which goes something like:

"Prove that 13^3 = 2197. Generalize your answer. (This is a horrible kind of problem that the author has tried to avoid)."

Questions:

1. How would you actually go about proving this ? (Direct multiplication is one way, another way could be using formula of (a+b)^3). Does the solution requires using some method that will allow us to make some kind of generalization ?

2. What is the generalization here ?

3. Why is this a horrible kind of problem ?

4. What are some other kind of similar horrible problems that you are aware of ?

Appreciate any answers.

P.S. I apologize if the statement of problem above makes it look like a homework problem, but its not. Request people to not tag this as a homework problem, so that more people can give answers.

Answer 1

I'd guess that he's alluding to perhaps proving it starting from just the Peano axioms. Then constructing the integers, and going on to formally show that 13^3 = 2197 is a natural, logical conclusion that flows from the definition of exponentiation.

We could generalize to show that given an a and b, there exists some integer c, that is a^b.

This is a horrible kind of a problem because most people find it uninteresting.

Similar sorts of problems can be found in a course on analysis (along with some greatly more interesting).