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recently, I asked a question concerning the number of solutions of a diophantine equation that used the rounding function. This question, however, dealt with a linear function, and I was wondering if the method or the answer could be generalized to include larger families of functions. I was trying to use the same technique given to me in the answer of that question to solve: $$ \lceil x(\ln (x \ln x))\rceil+ \lceil y(\ln (y \ln y))\rceil = N$$ However, I do not think the same method can be applied, given that this equation is non linear. I have tried but I got stuck in the spot with the minimums and maximums. Is there a function $f(N)$ that counts how many integer solutions this equation has?
Furthermore, is there a function $f(g(x),m,N)$ that counts the number of integer solutions of the following equation? $$\sum_{i=1}^m g(x_i)=N$$

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  • $\begingroup$ You know the study of integer or rational solutions of diophantine equations is the major motivating force behind much of modern number theory and arithmetic geometry right? After all, Fermat's last theorem was only settled in the last 20 years and studies solutions to $x^n + y^n = z^n$ (for some fixed $n$). Pretty much as soon as you start considering nonlinear equations you go from linear algebra (easy) to algebraic geometry (hard). I'm sure estimates exist for certain types of equations, but there is pretty much nothing that can be said of nonlinear equations in general $\endgroup$ – oxeimon Oct 30 '15 at 0:50
  • $\begingroup$ @oxeimon alright, so forget the last part of the question, what about the first? Is there an answer to the question regarding the first equation? $\endgroup$ – Guacho Perez Oct 30 '15 at 1:00
  • $\begingroup$ @WillJagy why does anyone want to know anything in mathematics? :) $\endgroup$ – Guacho Perez Oct 30 '15 at 2:04
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Of course there is such a function: you just defined it. The question is whether there is an algorithm for computing this function.

In general there is no way to tell whether a polynomial Diophantine equation (in several variables) has any solutions: see Hilbert's 10th Problem.

In your case, because the left side is greater than $x + y$ for $x, y \ge 3$, any solution must have $x + y \le N$, so there are only finitely many possibilities to try.

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  • $\begingroup$ And are there any methods to be used so I can find an explicit formula or even an algorithm? Maybe even properties of the function, i.e. bounds, asymptotes, etc.? Are there some books I can refer to? $\endgroup$ – Guacho Perez Oct 30 '15 at 2:05

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