Here I describe the sort of reference I'm after with a motivating example. I am **not** seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good references on solving equations of this sort.


I had an equation



> 
> $2 \lfloor a \rfloor\_{c} - a = d - c$
> 
> 
> 


with $a,c,d \in {\mathbb Z} $, and where $\lfloor n \rfloor\_{k}$ is my notation for $k \lfloor n/k \rfloor$ — essentially the floor function down to the nearest multiple of $k$.


I wished to solve for $a$. Now, I didn't know how to tackle this algebraically, as the usual technique of bringing the $a$'s together does not seem to be available. However, I had some notion of what form a solution was likely to take. After some guesswork and experimentation with *Maxima*, I found the solution:



> 
> $a = 2 \lceil d \rceil\_{c} - (c + d)$
> 
> 
> 


This appears to be correct, but I have not yet found a way to prove this — but that's not my question.


This approach is very unsatisfactory to me. I would much rather solve the problem algbraically.


**I'd like to know if there are any recommended references, either books or on-line, about techniques that can be used to solve equations involving the floor ($\lfloor \cdot \rfloor$), ceiling ($\lceil \cdot \rceil$), fraction-part, and similar functions, either in ${\mathbb Z}$, ${\mathbb Q}$ or ${\mathbb R}$.**


Beyond my particular equation of interest, I'd be interested to learn how to tackle this sort of equation more generally.


(In case you're interested why I was looking at this equation: I have recently encountered the remarkable Stern diatomic sequence. The equation in question is related to the successor function on ratios of consecutive terms; I wished to find the inverse function.)



References on techniques for solving equations with discontinuous functions such as floor and ceiling?
Joe Roberts provided the words for the calligraphed book Elementary Number Theory: A Problem Oriented Approach, which was printed in the 1970's. This book has a chapter on brackets, which in some of the number theory literature is an older name for one or both of the floor and ceiling functions. While not providing as focused a treatment of brackets, Concrete Mathematics, which was authored by Graham, Knuth, and Patashnik, also gives some service to the handling of floor and ceiling. With the bibliographies of those two books and a decent citation index, you may find more recent treatments. There may be other search terms to use, but I would start with "brackets +number theory -Lie" or something like that in a web search.


Gerhard "Is It Forty Years Already?" Paseman, 2013.04.15


