ProofWiki problem 28 Show that \(3 = \sqrt {1 + 2 \sqrt {1 + 3 \sqrt { 1 + \cdots} } }\).

Problem 28 is a more difficult problem and the model is completely unable to deal with it. It admits that problems involving nested radicals can be difficult and actually gives up after standard methods don't make any headway.

A consistent problem here is an inability to write down a correct expression for a recursive relation to describe the nested radical. GPT-4 seems to be convinced that the expression under each square root is the same, so that if we write the initial expression \(3 = \sqrt{A}\) then we also have \(3 = \sqrt{1 + 2\sqrt{A}}\) and \(3 = \sqrt{1 + 2\sqrt{1 + 3\sqrt{A}}}\), etc.

On subsequent attempts additional terms of the initial sequence were provided in the hope that it would pick up on the increasing sequence of constants that the square roots are multiplied by.

Whilst GPT-4 would confirm that it had noticed this pattern, it would always proceed ignoring this fact. On each generation, GPT-4 would finish off by noting it got the wrong answer and that this must be because it didn't take this increasing sequence of constants into account! It's as though GPT-4 only knows one way to handle nested radicals, and knowing that this won't work here, tries it anyway, inevitably getting the wrong answer.

To probe a little deeper, GPT-4 was instead prompted in a direction that might allow it to make partial progress. The hint was given to try peeling the expression on the right hand side one square root at a time, working backwards from the desired result that the full nested radical should have the value 3 to see if some pattern could be found in the values of the inner nested radicals.

It was easy to prompt it so that it heads in that direction but on every generation it made hopeless algebraic and numerical errors, once again illustrating that very often what holds it back is high school algebra rather than the depth of the mathematics.

As GPT-4 could not be coaxed into returning correct values for the sequence of inner nested radicals, the attempt to solve the problem using GPT-4 was abandoned.