Counting the number of solutions to the Erdős-Straus equation on unit fractions (Q2852274)

For any positive integer \(n\), let \(f(n)\) denote the number of positive integer solutions \((x,y,z)\) to the Diophantine equation NEWLINE\[NEWLINE {4 \over n}= {1\over x}+{1\over y}+{1\over z}. \tag{1} NEWLINE\]NEWLINE The \textit{Erdős-Strauss} conjecture says that \(f(n)>0\) for every \(n\geq 2\) and there is a rather large literature around the above equation and conjecture. The announcements of the various results included in this long and interesting paper require a rather large number of notions and technical definitions to be included in this review. Nevertheless, a corollary of one of the main results of the paper is the asymptotic formula NEWLINE\[NEWLINE N\log^2N\ll \sum_{p\leq N} f(p) \ll N\log^2 N\log\log N, NEWLINE\]NEWLINE where the parameter \(p\) means ``prime'' and the implied constant in the notation \(x\ll y\) (equivalent to \(x=O(y)\)) is absolute. Note that the asymptotic study of \(f(n), n\in\mathbb{N}\) is reduced to that of \(f(p), p\) prime.NEWLINENEWLINEA basic methodological tool of the authors' s approach is (following a number of earlier papers) to divide the solutions to (1) into two classes and to focus to each class separately: The solutions \((x,y,z)\) of \textit{Type I} are those for which \(n|x\) and \(\gcd(n,yz)=1\); their number is denoted by \(f_{\mathrm{I}}(n)\). The \textit{Type II} solutions are those for which \(\gcd(n,x)=1\) and \(n\) divides both \(y\) and \(z\); their number is denoted by \(f_{\mathrm{II}}(n)\). By permuting \(x,y,z\) we see that \(f(n)\geq 3f_{\mathrm{I}}(n)+3f_{\mathrm{II}}(n)\) for all \(n>1\). Also, it is not difficult to see that, if \(p\) is an odd prime, then \(f(p) = 3f_{\mathrm{I}}(p)+3f_{\mathrm{II}}(p)\).NEWLINENEWLINEThe authors treat also the polynomial solutions to the equation (1) and classify all solvable primitive congruences. More specifically, let \(q\) be a given positive integer and \(r\) a positive integer relatively prime to \(q\) (in other words, \(r\bmod q\) is a \textit{primitive class} \(\bmod\,q\)). By definition, \textit{the class \(n=r\bmod q\) is solvable by polynomials}, if there exist polynomials \(P_i(n)\) (\(i=1,2,3\)) which take positive integer values for all sufficiently large \(n=r\bmod q\) (in particular, this implies that the polynomials \(P_i(n)\) have rational coefficients), such that NEWLINE\[NEWLINE {4 \over n}= {1\over P_1(n)}+{1\over P_2(n)}+{1\over P_3(n)}. NEWLINE\]NEWLINE The last section of the paper (save the appendix) is devoted to the more general equation NEWLINE\[NEWLINE {m\over n}={1\over t_{1}}+{1\over t_{2}}+\cdots +{1\over t_{k}}. NEWLINE\]NEWLINE An appendix in the end contains various useful number-theoretical results. Finally an extended bibliography of 86 items is included.NEWLINENEWLINEEach section of the paper is very neatly written; what this reviewer misses is a plain description of the structure of the paper in the introduction.