This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in which one of the main texts is Hofstadter's book Gödel, Escher, Bach. In one of its dialogs, the Tortoise explains that he has found a counterexample to Fermat's Last Theorem, and the exponent $n$ in question is the unique positive integer not appearing in the continued fraction expansion of $\pi$ — and this is enough information for you also to find it.
This is funniest if the following were true at the time of GEB's writing (1978):
(a) it was widely believed, but not known, that every positive integer appears somewhere in the continued fraction expansion of $\pi$;
(b) proving this was far beyond the technology of the time, and even saying anything useful about which numbers do and don't appear (beyond calculating the first few numbers that do appear) looks interminable.
Based on what little I know about similar problems, my guess is that (a) and (b) are both true, and remain true today. Evidence for this impression is OEIS sequence A225802. But this is far from my area of expertise. Thus, my questions are:
- Were (a) and (b) true in 1978?
- Do they remain true today?