Unit fractions with shifted prime denominators

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash\{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

The study of decompositions of rational numbers into sums of distinct unit fractions (often called 'Egyptian fractions') is one of the oldest topics in number theory (see [2] for further background and many related problems on Egyptian fractions).It is elementary to prove that such a decomposition is always possible, for instance by using a greedy algorithm.In this paper we explore a natural variant that imposes restrictions on the denominators in these decompositions.
Question 1.For which A ⊆ N is it true that every positive rational number can be written as n∈B 1 n for some finite B ⊂ A? A trivial necessary condition is that the set contains multiples of every prime; for example, the set of all odd numbers does not have this property (it cannot represent 1 2 ).The condition n∈A 1 n = ∞ is also clearly necessary, but not sufficient -indeed, it is easy to see that there is no solution to 1 = p∈A 1 p where A is any finite set of primes, even though p≤N 1 p ∼ log log N .For sets A with no such trivial obstructions it is reasonable to speculate that an Egyptian fraction decomposition with denominators restricted to A always exists.An early seminal paper on this topic is by Graham [6], who proved a general result that implies, for example, that such a decomposition always exists when A is the set of all primes and squares.Motivated by a conjecture of Sun [12,Conjecture 4.1], Eppstein [4] developed an alternative elementary method, which implies such a decomposition always exists when A is the set of 'practical numbers' (those n such that all m ≤ n can be written as the sum of distinct divisors of n).
A variant of Question 1 can be asked even when there are trivial obstructions.For example, Graham [6] has shown that every rational number x can be written as the sum of distinct unit fractions with square denominators, subject to the obvious necessary condition that x ∈ [0, ), and every rational number x with square-free denominator can be written as the sum of distinct unit fractions with square-free denominators.
A natural candidate of number theoretic interest, for which there exist no obvious obstructions to any rational decomposition, and for which the methods of [6] and [4] are not applicable, is the set of shifted primes {p − 1 : p prime}.That such a restricted Egyptian fraction decomposition always exists was conjectured by Sun [12,Conjecture 4.1] (see also [13,Conjecture 8.17] and [11] for some numerical data).In this paper we use the method of [1] to prove this conjecture: any positive rational x > 0 has a solution (indeed, infinitely many) to We also prove a similar result with denominators p i − h for any (fixed) h = 0, although for |h| > 1 there are some trivial congruence obstructions -for example, since no subset of {p + 2 : p prime} has lowest common multiple divisible by 8 the fraction 1  8 cannot be represented as the sum of distinct unit fractions of the shape 1 p+2 .We deduce this existence result from the following more general result, showing that any shifted set of primes, all divisible by q, of 'positive upper relative logarithmic density' contains a decomposition of 1 q .(Recall that p≤N 1 p ∼ log log N , and so it is natural to consider p≤N p∈A 1 p divided by log log N as a measure of the size of A.) Theorem 1.Let h ∈ Z\{0} and q ≥ 1 be such that (|h| , q) = 1.If A is a set of primes congruent to h (mod q) such that lim sup then there exists a finite S ⊂ A such that A simple application of partial summation produces the following version with (relative) upper logarithmic density replaced by (relative) lower density.
Corollary 1.Let h ∈ Z\{0} and q ≥ 1 be such that (|h| , q) = 1.If A is a set of primes congruent to h (mod q) with positive relative lower density, that is, then there exists a finite S ⊂ A such that We remark that (unlike the statement for unrestricted sets of integers, see [1, Theorem 2]) the stronger version of Corollary 1 with the lim inf N →∞ replaced by lim sup N →∞ is false -for example, if where N k = 2 k C for some large absolute constant C > 0 then n∈A 1 n < 1, and hence certainly we cannot find a finite S ⊂ A such that n∈S 1 n = 1, and yet lim sup We now show how Theorem 1 implies the headline result: any positive rational number (subject to the necessary congruence conditions) can be written as the sum of distinct unit fractions with shifted prime denominators.
Corollary 2. Let h ∈ Z and x = r/q ∈ Q >0 be such that (|h| , q) = 1.There are distinct primes p 1 , . . ., p k such that Proof.By Dirichlet's theorem (see for example [10, Corollary 4.12]) if A is the set of primes congruent to h (mod q), then Trivially the same must hold for A\B, for any finite set B. In particular by r repeated applications of Theorem 1 (first to A, then A\S 1 , and so on) we can find We prove Theorem 1 with an application of the author's earlier work [1] (which in turn is a stronger form of an argument of Croot [3]).Loosely speaking, the main result of [1] shows that we can solve 1 = every n ∈ A is 'friable' (or 'smooth'), in that if a prime power q divides n then q ≤ n1−δ(n) for some 0 < δ(n) = o(1), (3) every n ∈ A has 'small divisors', and (4) every n ∈ A has ≈ log log n many distinct prime divisors.
To prove Theorem 1, therefore, it suffices to show that the set { p−h q : p ∈ A} has these properties.Fortunately, there has been a great deal of study of the arithmetic properties of shifted primes, and so using classical techniques from analytic number theory we are able to find a large subset of our original set A satisfying all four properties.
For experts in analytic number theory we add that in establishing the necessary number theoretic facts about shifted primes we have followed the simplest path, forgoing many of the more elaborate refinements possible.The main observation of this paper is that the inputs required to the method of [1] are mild enough to be provable for the shifted primes using (a crude form of) existing technology.
To minimise technicalities we have proved only a qualitative form of Theorem 1.In principle a (very weak) quantitative version could be proved with the same methods, along similar lines to [1, Theorem 3], but this would complicate the presentation significantly.
Finally, the methods and main results of [1] have now been formally verified using the Lean proof assistant, in joint work with Bhavik Mehta. 1 This formalisation has not been extended to the present work, but since the proof of Theorem 1 uses the main result of [1] as its primary ingredient (combined with classical number theory) it can be viewed as 'partially formally verified'.
In Section 1 we prove Theorem 1 assuming certain number theoretic lemmas.In Section 2 we prove these lemmas.
Acknowledgements.The author is funded by a Royal Society University Research Fellowship.We would like to thank Greg Martin for a helpful conversation about friable values of shifted primes and remarks on an earlier version of this paper.

Proof of Theorem 1
Our main tool is the following slight variant of [1, Proposition 1] (which is identical to the below except that the exponent of c is replaced by 1/ log log N ).
) every n ∈ A is divisible by some d 1 and d 2 where y ≤ d 1 and 4d 1 ≤ d 2 ≤ z, (3) every prime power q dividing some n ∈ A satisfies q ≤ N 1−4c , and (4) every n ∈ A satisfies There is some S ⊆ A such that n∈S Proof.The proof is identical to that of [1, Proposition 1], except that in the final part of the proof we choose M = N 1−c .Observe that the inputs to that proof, namely [1, Proposition 2, Proposition 3, and Lemma 7], are valid for any M ∈ (N 3/4 , N ).It remains to check the 'friable' hypothesis, for which we require that if n ∈ A and q is a prime power with q | n then, for some small absolute constant c > 0, For N sufficiently large (depending only on c) the right-hand side is > N 1−4c , and so hypothesis (3) suffices.
It is convenient to recast this in a slightly different form.
Proposition 2. Let δ, ǫ > 0 and suppose y is sufficiently large depending on δ and ǫ, and y ≤ w ≤ z.If N is sufficiently large (depending on δ, ǫ, y, w, z) and n is divisible by some d 2 ∈ [4w, z), and Proof.For i ≥ 0 let N i = N (1−ǫ/4) i , and let A i = A ∩ (N i+1 , N i ].Note that N i < 2 for i ≥ C log log N , where C is some sufficiently large constant depending only on ǫ.Since n≤log log N 1 n ≪ log log log N it follows by the pigeonhole principle that there must exist some i such that with A ′ = A i and N ′ = N i ≫ log log N we have It suffices to verify that the assumptions of Proposition 1 are satisfied by A ′ , with c = ǫ/4.We have already verified the first assumption (assuming y and N are sufficiently large; note that since N ′ ≫ log log N this ensures that N ′ is also sufficiently large).The second assumption of Proposition 1 is ensured by conditions (3) and ( 4).
For the third assumption, note that by condition (1) if n ∈ A ′ is divisible by a prime power q then q ≤ n 1−ǫ ≤ (N ′ ) 1−ǫ as required.Finally the fourth assumption follows from condition (2) and noting that for all n ∈ [(N ′ ) 1−ǫ/4 , N ′ ] we have log log n = log log N ′ + O ǫ (1), and the O ǫ (1) term is ≤ log log N ′ /500, say, provided we take N sufficiently large.
To prove Theorem 1 we want to apply Proposition 2 to B = { p−h q : p ∈ A}.To verify the hypotheses we will require the following number-theoretic lemmas.We were unable to find these exact statements in the literature, so have included proofs in the following section, but the proofs are all elementary and cover well-trodden ground.
Lemma 1.For any ǫ > 0 and h ∈ Z\{0} the relative density of primes p such that n = p − h is divisible by a prime power q > n 1−ǫ is O h (ǫ).
Lemma 2. For any δ > 0 and h ∈ Z\{0} the relative density of primes p such that Lemma 3.For any h ∈ Z\{0}, if 4 ≤ y < z the relative density of primes p such that n = p − h is not divisible by any primes q ∈ [y, z] is O h (log y/ log z).
We will now show how these lemmas, combined with Proposition 2, imply Theorem 1.
Proof of Theorem 1.By assumption there is some δ > 0 and infinitely many N such that Let B = { p−h q : p ∈ A} ⊂ N, so that there must exist infinitely many N such that Let ǫ = cδ where c > 0 is some small absolute constant to be determined later.Let y be sufficiently large in terms of δ (so that Proposition 2 can apply) and w ≤ z be determined shortly, and let B ′ ⊆ B be the set of those n ∈ B such that Fix such an N and let All of the conditions from Proposition 2 are now satisfied for B ′′ , and hence there exists some S 1 ⊆ B ′′ and d 1 ≤ z such that n∈S1 1 n = 1 d1 .We now apply Proposition 2 again to B ′′ \S 1 , and continue this process k = ⌈z⌉ 2 many times, producing some disjoint S 1 , . . ., S k and associated d 1 , . . ., d k ≤ z where Notice that the conditions of Proposition 2 remain satisfied for each B ′′ \ ∪ i≤j S i for j ≤ k, since assuming N is sufficiently large, since z depends on δ only.By the pigeonhole principle there must exist some d ≤ z and i 1 , . . ., i d such that as required.

Number theoretic ingredients
It remains to prove Lemmas 1, 2, and 3, which we will do in turn.

Friability of shifted primes.
There has been a great deal of work on shifted primes with only small prime divisors.Often the focus is on an existence result, finding the smallest possible δ > 0 such that there exist infinitely many shifted primes p − 1 with no prime divisors > p δ .We refer to [9] for recent progress on this and references to earlier work.Our focus is a little different: we are content with a very high friability threshold, but we need to show that almost all shifted primes are this friable.For the regime of friability that we are interested even the original elementary methods of Erdős [5] suffice.
Proof of Lemma 1.This is only a slight generalisation of [5, Lemma 4].It suffices to show that, for all ǫ > 0 and large N , the number of p ≤ N such that p − h is divisible by some prime power q with q > N 1−ǫ is We first note that trivially for any q the number of p ≤ N such that p−h is divisible by q is certainly O h (N/q), and hence the count of those p − h divisible by some non-prime prime power q > N 1−ǫ is for all large N .It remains to bound the count of those p ≤ N such that p − h is divisible by some prime q > N 1−ǫ .Such p−h we can write uniquely (assuming N is large enough depending on h) as p − h = qa for some a ≤ 2N ǫ and q > N 1−ǫ prime.A simple application of Selberg's sieve (for example [8,Theorem 3.12]) yields that, for any fixed a ≥ 1 and h = 0 the number of primes q ≤ x such that aq + h is also prime is Summing over all a ≤ 2N ǫ the total count is ≪ ǫ N log N as required, using the fact that a≤M 1 φ(a) ≪ log M .2.2.Number of prime divisors of shifted primes.We need to know that ω(n) ∼ log log n for almost all n ∈ {p − h : p prime}.This is in fact the typical behaviour of ω(n) for a generic integer n, and we expect the same behaviour when restricting n to the random-like sequence of shifted primes.Indeed, just like ω(n) itself, ω(p−h) satisfies an Erdős-Kac theorem, that is, ω(p−h) behaves like a normal distribution with mean log log(p − h) and standard deviation log log(p − h).This was established by Halberstam [7], although a simple variance bound suffices for our application here.For the required sieve input we will use the following classic result on the distribution of primes within arithmetic progressions (which is proved, for example, as [10,Corollary 11.21]).Recall that π(N ; d, h) is the number of primes p ≤ N such that p ≡ h (mod d).
Theorem 2 (Siegel-Walfisz).There is a constant c > 0 such that for all h ∈ Z and The conclusion now follows provided we choose N large enough so that z ≪ √ log log N , say, and using Mertens' estimate that p≤w (1 − 1/p) ≍ 1 log w .

n∈A 1 n
≥ δ log log N , then there exists S ⊆ A such that n∈S 1 n = 1/d for some d ≤ z.

Proof of Lemma 2 .
It suffices to show that, for all δ > 0 and large N , if A is the set of p ≤ N such that |ω(p − h) − log log(p − h)| > δ log log(p − h), then |A| ≪ N (log N )(log log N ).Let A 1 = A ∩ [1, N1/2 ] and A 2 = A\A 1 .We can trivially bound |A 1 | ≪ N 1/2 , and for p ∈ A 2 we have log log(p − h) = log log N + O(1), whence for large enough N if p ∈ A 2 we have |ω(p − h) − log log N | > δ 2 log log N. By [7, Theorem 3], however, we have p≤N|ω(p − h) − log log N | 2 ≪ π(N ) log log N,and hence |A 2 | (log log N ) 2 ≪ δ π(N ) log log N, and the result now follows from Chebyshev's estimate π(N ) ≪ N/ log N .2.3.Shifted primes with small divisors.For Lemma 3 we need to show that there are few shifted primes remaining after we remove all multiples of primes p ∈ [y, z], which is a classic upper bound sieve problem.Since the information we require is very weak even the simplest sieve suffices: the following is proved as[8, Theorem 1.1].Lemma 4 (Sieve of Eratosthenes-Legendre).Let A be a finite set of integers and P a finite set of primes.Let z ≥ 2 and P (z) = p∈P p<z p. Suppose that f (d) is a multiplicative function and X > 1 is such that for all d | P (z) we have |A d | = f (d)X + R d .Then #{n ∈ A : (n, P (z)) = 1} ≪ X p∈P p<z (1 − f (p)) + d|P (z) |R d | .