PODSYPANIN'S PAPER ON THE LENGTH OF THE PERIOD OF A QUADRATIC IRRATIONAL
John P. Robertson · 2007
where ` is the length of the period of the continued fraction expansion of a real quadratic irrational with discriminant ∆ (not necessarily a fundamental discriminant), is well known and correct. But, there are a number of errors in [11], including that the proofs of both lemmas are wrong. Fortunately, all of the errors are easily fixed. I would guess that both lemmas were well-known long before [11] was published. Here’s the list of errors: (1) His proof of his Lemma 1 is wrong. (2) His statement of his Lemma 2 is not quite correct, and his proof is wrong. (3) The misstatement of Lemma 2 carries through to a misstatement of his theorem. (4) An allusion to a reference for a result is potentially misleading. (5) There are at least two minor typographical errors. This note discusses these matters. For Lemma 1 we tell where you can find a proof, and for Lemma 2 we supply a proof along the lines of the proof in [11]. We fill in some other small gaps in the paper. We end with a brief restatement of his main argument in the section titled “Recap.”