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 of solutions to the Diophantine equation

with positive integers. The Erdös-Straus conjecture asserts that for all . Since for all positive integers , it suffices to show that for all primes .

We single out two special types of solutions: *Type I* solutions, in which is divisible by and are coprime to , and *Type II* solutions, in which is coprime to and are divisible by . Let denote the number of Type I and Type II solutions respectively. For any , one has

with equality when is an odd primes . Thus, to prove the Erdös-Strauss conjecture, it suffices to show that at least one of , is positive whenever is an odd prime.

Our first main results are the asymptotics

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

and

The double logarithmic factor in the upper bound for 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The methods are similar to those in the previous paper (which were also independently discovered in unpublished work of Elsholtz and Heath-Brown), but with the additional input of the Erdös divisor bound on expressions of the form for polynomials , discussed in this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 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

thus, we see that for typical , that most solutions to the Erdös-Straus equation are not of Type I or Type II, in contrast to the case when 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 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 and , namely

and

which should be compared to a recent previous bound of Browning and Elsholtz. In the other direction, we show that for infinitely many , and for almost all primes . 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 whenever one of the following congruence-type conditions is satisfied:

- , where are such that .
- and , where are such that .
- and , where are such that .
- or , where are such that and .
- , where are such that .
- , where are such that .

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

Finally, we begin a study of the more general equation

where are fixed. We can obtain a partial analogue of our main bounds for the case, namely that

and

were denotes the number of solutions to (2) which are of “Type II” in the sense that are all divisible by . However, we do not believe our bounds to be sharp in the large regime, though it does show that the expected number of solutions to (2) should grow rapidly in .

## 10 comments

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20 August, 2011 at 10:17 pm

mixedmathIs there anything known about the (perhaps easier) case when we look for solutions to ? I am trying to think whether this makes the idea easier or not.

21 August, 2011 at 8:50 am

Terence TaoIn our paper we have some lower bounds on the average number of solutions in this case (which we call the k=4 case), but we were unable to match them with good upper bounds. There are of course more solutions in the k=4 case than the k=3 case (and in particular, there is obviously at least one solution for each n) but by the same token, they are harder to parameterise efficiently.

13 September, 2011 at 10:34 pm

petegIn Lemma 2.7 of the paper and its proof, bounds are given for some of the parameters of Type I and II solutions. In particular, it is proved that, for Type I solutions, , where is the smallest value among , , and . It is then stated that “The constants in the bounds here could be improved slightly, but such improvements will not be of importance in our applications.”

I believe I can show that the upper bound on in the Type I solution case can be improved to , unless , in which . I’m not sure, maybe this is what the authors had in mind when it was said that the bounds could be improved slightly. Anyway, for what it’s worth, here is my argument. I use the notation from the paper, Lemma 2.7.

First, observe that , since and . Next, observe that . This follows by starting from (2.3) in the paper, , and, since , it follows that . Next, observe that . This follows from , by substituting . Next, observe that . This follows from , by substituting . Thus the solution .

Assume now that . Then , say where . So .

But since , . If , then since , , a contradiction. So assume . Then , so . But , so this means . This means . But since , this means that , and, since , it must be that , so = 1 or 2. If , then , which is clearly impossible. If , then and , so , and . Since , this case can only happen when .

20 September, 2011 at 2:48 am

Diophantine sets and the integers | cartesian product[…] Counting the number of solutions to the Erdös-Straus equation on unit fractions (terrytao.wordpress.com) […]

20 January, 2012 at 7:04 pm

daviecan i have the list of erdos-straus conjecture expressing rational number 4/n a sum of three unit fractions 1/x+1/y+1/z???email me at my account atillodavid@yahoo.com…tnx a lot

23 August, 2014 at 9:11 am

ibrahimaeygueDear all

RSA numbers linked to the pythagorean triples associated with Erdös-Straus decompositions…. N = 2099999959 is factorised in less than 10 / 100th of a second almost “instantly” The general characterization is given by :

– p = s*N*r/((2*a*s – r*N)*( s+r+t))

– q = ((2*a*s – r*N)*( s+r+t))/(s*r)

and/or

– p = s*N*r/((2*a*s – r*N)*( s-r+t))

– q = ((2*a*s – r*N)*( s-r+t))/(s*r)

Best regards

17 October, 2016 at 9:02 am

Ben thurstonI found these I hadn’t seen anywhere else: http://imgur.com/a/mAlIR

17 October, 2016 at 9:17 am

Ben thurstonthey make a little more sense factoring a term I was leaving expanded…http://imgur.com/a/nDuao

18 October, 2016 at 12:54 am

wlodPerhaps, in the old days, many people got the result for all $n$ but for $12\cdot m+1$ case. This follows from the three simplest cases: $4\cdot m-1$, and $3\cdot m-1$, and $8\cdot m-3$.

28 August, 2019 at 1:59 am

siplebriceI was toying with Integer Partitions of 4n into exactly 3 parts restricted to the divisors of pn. I’m not sure how to address that restriction combinatorially, but if a+b+c=4n and a, b, & c each divide pn, it’s done.