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Here we denote $[a_0,...,a_n]$ as the continued fraction of some rational number. If I take $p/q=[a_0,a_1,...,a_n]$ to $p'/q'=[a_0+1,a_1+1,...,a_n+1]$, are there any nice properties I can say about $p'/q'$?

Ѕᴀᴀᴅ
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continuedFractionQs
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    Personally, unknown, but fascinating question. If I was attacking this question, my first stop would be Chapter 1 (only) of this book. Then, my second stop would be this book. – user2661923 Oct 13 '22 at 10:26
  • I find the question interesting as well. What kind of properties are you interested in: Bounds with respect to $p/q$, size of $q'$ relative to $q$, factorization properties of $p$ and $q$? With regards to the latter, I find it hard to expect there to be any nice results on the prime factorization of $p'$ and $q'$ - the examples I have computed seem to behave wildly in that regard. I have written a short Python script for generating examples, I can share if you're interested. – SomeCallMeTim Oct 13 '22 at 11:38
  • When all $a_i$ are identical, you can compare them using this: https://math.stackexchange.com/questions/4453134/expressing-the-continued-fraction-k-dots-k-as-a-closed-form (Note: since you are dealing with plus signs, use polynomials of the first kind instead, with a plus sign inside the square root) – Collag3n Oct 14 '22 at 08:59

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We have $p' = p + q$ and $q' = q$. This is because, in a continued fraction, the $n$-th term is $a_n = \lfloor \frac{p}{q} \rfloor$, the greatest integer less than or equal to $\frac{p}{q}$. But we have $$ \frac{p}{q} = a_0 \frac{1}{1} + \frac{1}{a_1 \frac{1}{1} + \frac{1}{\ddots + \frac{1}{a_n}}} $$ so $$ \frac{p + 1}{q + 1} = a_0 \frac{1}{1} + \frac{1}{a_1 \frac{1}{1} + \frac{1}{\ddots + \frac{1}{a_n + \frac{1}{1}}}} = \frac{p'}{q'} $$ where $p' = p + q$ and $q' = q$.

Amirreza Hashemi
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