Christian Elsholtz and I have recently finished our joint paper “Counting the number of solutions to the Erdös-Straus equation on unit fractions“, submitted to the Journal of the Australian Mathematical Society. This supercedes my previous paper on the subject, by obtaining stronger and more general results. (The paper is currently in the process of being resubmitted to the arXiv, and should appear at this link within a few days.)

As with the previous paper, the main object of study is the number ${f(n)}$ of solutions to the Diophantine equation

$\displaystyle \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \ \ \ \ \ (1)$

with ${x,y,z}$ positive integers. The Erdös-Straus conjecture asserts that ${f(n)>0}$ for all ${n>1}$. Since ${f(nm) \geq f(n)}$ for all positive integers ${n,m}$, it suffices to show that ${f(p)>0}$ for all primes ${p}$.

We single out two special types of solutions: Type I solutions, in which ${x}$ is divisible by ${n}$ and ${y,z}$ are coprime to ${n}$, and Type II solutions, in which ${x}$ is coprime to ${n}$ and ${y,z}$ are divisible by ${n}$. Let ${f_I(n), f_{II}(n)}$ denote the number of Type I and Type II solutions respectively. For any ${n}$, one has

$\displaystyle f(n) \geq 3 f_I(n) + 3 f_{II}(n),$

with equality when ${n}$ is an odd primes ${p}$. Thus, to prove the Erdös-Strauss conjecture, it suffices to show that at least one of ${f_I(p)}$, ${f_{II}(p)}$ is positive whenever ${p}$ is an odd prime.

Our first main results are the asymptotics

$\displaystyle N \log^3 N \ll \sum_{n \leq N} f_I(n) \ll N \log^3 N$

$\displaystyle N \log^3 N \ll \sum_{n \leq N} f_{II}(n) \ll N \log^3 N$

$\displaystyle N \log^2 N \ll \sum_{p \leq N} f_I(p) \ll N \log^2 N \log\log N$

$\displaystyle N \log^2 N \ll \sum_{p \leq N} f_{II}(p) \ll N \log^2 N.$

This improves upon the results in the previous paper, which only established

$\displaystyle N \log^2 N \ll \sum_{p \leq N} f_I(p) \ll N \exp(O( \frac{\log x}{\log\log x} ))$

and

$\displaystyle N \log^2 N \ll \sum_{p \leq N} f_{II}(p) \ll N \log^2 N \log \log N.$

The double logarithmic factor in the upper bound for ${\sum_{p \leq N} f_I(p)}$ is artificial (arising from the inefficiency in the Brun-Titchmarsh inequality on very short progressions) but we do not know how to remove it.

The methods are similar to those in the previous paper (which were also independently discovered in unpublished work of Elsholtz and Heath-Brown), but with the additional input of the Erdös divisor bound on expressions of the form ${\sum_{n \leq N} \tau(P(n))}$ for polynomials ${P}$, discussed in this recent blog post. (Actually, we need to tighten Erdös’ bound somewhat, to obtain some uniformity in the bounds even as the coefficients of ${P}$ become large, but this turns out to be achievable by going through the original arguments of Erdös more carefully.)

We also note an observation of Heath-Brown, that in our notation gives the lower bound

$\displaystyle N \log^6 N \ll \sum_{n \leq N} f(n);$

thus, we see that for typical ${n}$, that most solutions to the Erdös-Straus equation are not of Type I or Type II, in contrast to the case when ${n}$ is prime.

We also have a number other new results. We find a way to systematically unify all the previously known parameterisations of solutions to the Erdös-Straus equation, by lifting the Cayley-type surface ${\{ (x,y,z): \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \}}$ to a certain three-dimensional variety in six-dimensional affine space, in such a way that integer points in the former arise from integer points in the latter. Each of the previously known characterisations of solutions then corresponds to a different choice of coordinates on this variety. (This point of view was also adopted in a paper of Heath-Brown, who interprets this lifted variety as the universal torsor of the Cayley surface.) By optimising between these parameterisations and exploiting the divisor bound, we obtain some bounds on the worst-case behaviour of ${f_I(n)}$ and ${f_{II}(n)}$, namely

$\displaystyle f_I(n) \ll n^{3/5 + O(1/\log \log n)}$

and

$\displaystyle f_{II}(n) \ll n^{2/5 + O(1/\log \log n)},$

which should be compared to a recent previous bound ${f(n) \ll n^{2/3 + O(1/\log \log n)}}$ of Browning and Elsholtz. In the other direction, we show that ${f(n) \gg n^{(3+o(1))/\log\log n}}$ for infinitely many ${n}$, and ${f(p) \gg \log^{\frac{\log 3}{2}-o(1)} p}$ for almost all primes ${p}$. Here, the main tools are some bounds for the representation of a rational as a sum of two unit fractions in the above-mentioned work of Browning and Elsholtz, and also the Turán-Kubilius inequality.

We also completely classify all the congruence classes that can be solved by polynomials, completing the partial list discussed in the previous post. Specifically, the Erdös-Straus conjecture is true for ${n}$ whenever one of the following congruence-type conditions is satisfied:

1. ${n = -f \mod 4ad}$, where ${a,d,f \in {\bf N}}$ are such that ${f|4a^2 d+1}$.
2. ${n = -f \mod 4ac}$ and ${n = -\frac{c}{a} \mod f}$, where ${a,c,f \in {\bf N}}$ are such that ${(4ac,f)=1}$.
3. ${n = -f \mod 4cd}$ and ${n^2 = -4c^2d \mod f}$, where ${c,d,f \in {\bf N}}$ are such that ${(4cd,f)=1}$.
4. ${n = -\frac{1}{e} \mod 4ab}$ or ${n = -e \mod 4ab}$, where ${a,b,e \in {\bf N}}$ are such that ${e|a+b}$ and ${(e,4ab)=1}$.
5. ${n = -4a^2d \mod f}$, where ${a,d,f \in {\bf N}}$ are such that ${4ad|f+1}$.
6. ${n = -4a^2d-e \mod 4ade}$, where ${a,d,e \in {\bf N}}$ are such that ${(4ad,e)=1}$.

In principle, this suggests a way to extend the existing verification of the Erdös-Straus conjecture beyond the current range of ${10^{14}}$ by collecting all congruences to small moduli (e.g. up to ${10^6}$), and then using this to sieve out the primes up to a given size.

Finally, we begin a study of the more general equation

$\displaystyle \frac{m}{n} = \frac{1}{n_1}+\ldots+\frac{1}{n_k} \ \ \ \ \ (2)$

where ${m > k \geq 3}$ are fixed. We can obtain a partial analogue of our main bounds for the ${m=4,k=3}$ case, namely that

$\displaystyle \sum_{n \leq N} f_{m,k,II}(n) \gg N \log^{2^{k-1}-1} N$

and

$\displaystyle \sum_{p \leq N} f_{m,k,II}(p) \gg N \log^{2^{k-1}-2} N / \log\log N$

were ${f_{m,k,II}(n)}$ denotes the number of solutions to (2) which are of “Type II” in the sense that ${n_2,\ldots,n_k}$ are all divisible by ${n}$. However, we do not believe our bounds to be sharp in the large ${k}$ regime, though it does show that the expected number of solutions to (2) should grow rapidly in ${k}$.