A NOTE ON A CONJECTURE OF ERDŐS AND ROSENFELD

Jorge Jiménez-Urroz Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, ESPAÑA.

Abstract. We prove a special case of a conjecture of Erdős and Rosenfeld regarding factor-difference sets of integers.

1. Introduction

Let n be an integer. In [1], P. Erdős and M. Rosenfeld defined the set D(n), the factor-difference set of n, as

D(n) = {d : d = |a − b|, n = ab}

and made the following conjecture: Conjecture. For every positive integer k there are k integers N_{1} < ⋯ < N_{k} such that | ⋂_{i=1}^{k} D(N_{i})| ≥ k.

In the paper quoted, they proved the conjecture for k = 2. In fact they prove: Theorem 1. For every positive integer k there are k integers N_{1} < ⋯ < N_{k} such that | ⋂_{i=1}^{k} D(N_{i})| ≥ 2.

In this paper we will prove the result analogous to Theorem 1 when k = 3. Theorem. For every positive integer k there are k integers N_{1} < ⋯ < N_{k} such that | ⋂_{i=1}^{k} D(N_{i})| ≥ 3.

To prove this theorem we study elliptic curves with full 2-torsion over ℚ. It turns out that any family of k integers as in the theorem, is obtained from k corresponding rational points on an elliptic curve with positive rank.

2. Proof of the theorem

Typeset by AMS-TEX

If N has d as a difference, 4N will have 2d in its factor-difference set. Therefore, without loss of generality we assume that all differences are even. Furthermore, 2d ∈ D(N) if and only if there is an integer x such that (x + d)(x − d) = x^{2} − d^{2} = N. Hence, if we want N to have three differences, say 2d_{1}, 2d_{2}, 2d_{3}, we have to find solutions to

(1)  N = x^{2} − d_{1}^{2} = y^{2} − d_{2}^{2} = z^{2} − d_{3}^{2}.

If A = d_{1}^{2} − d_{2}^{2}, B = d_{1}^{2} − d_{3}^{2}, then (1) is equivalent to the system

(2)  x^{2} − y^{2} =A
x^{2} − z^{2} =B.

To prove the theorem, we have to find three solutions (x, y, z) ∈ ℤ^{3} to (2), where A and B are integers of the particular form mentioned above.

This is the same as finding four integer solutions to (2), for any A, B ∈ ℤ. One of the solutions will define the differences, and the rest will give us three different N having those differences in common.

Further, we note that for any fixed n, the existence of n integer solutions to (2) is equivalent, by clearing denominators, to finding n solutions in ℚ^{3} for some A, B ∈ ℚ.

Let us call x^{2} = ξ. Then, (2) will be

ξ = x^{2},
ξ − A = y^{2},
ξ − B = z^{2}.

The following well known lemma, (see [2], (10.1.4)), will give us an equivalent condition to our initial problem, in terms of solutions of an elliptic curve.

Lemma. Let E(A, B) be the elliptic curve given by

E(A, B) := {(x, y) | y^{2} = x(x − A)(x − B)},

where A, B ∈ ℚ. A point P = (x, y), P ∈ E(A, B) with rational coordinates is in 2E[ℚ] if and only if x, x − A, x − B are all squares in ℚ.

So, in order to prove the theorem it will be enough to show that for some A, B ∈ ℚ the elliptic curve E(A, B) has positive rank.

Using the MAPLE package for elliptic curves created by Ian Connell , APECS, one finds, for 1 ≤ A, B ≤ 10, the pairs (A, B) ∈ {(1, 7), (2, 5), (2, 10), (3, 9), (3, 10), (4, 6), (5, 8), (5, 10), (6, 7), (6, 10), (9, 10)} such that E(A, B) has positive rank.

Remark.

A necessary condition to establish Theorem 1 for higher k, is to find curves C/ℚ with affine equation, y^{2} = f(x), where deg(f(x)) = 2k − 2, with “many” rational points. Example.

Considering E(4, 6) one finds 5 integers having 3 differences in common. Indeed, the integers

d1 = 993243542205992332488058754263122078071239561123822165112

d2 = 573704451935072432558657786561844237570109187841809114488
d3 = 20948274110993405271036818636118416362912097297024759288

and

N1 =416252465020561454407395364419197194144012042895155178800638

062718880469015509289397065832372310217474579962921088

N2 =150663274529569282064072003646177602927377024496937825623312

349879413129373761483685425020517086615471768268066864

N3 =552553293023056676163820319615821175151317152990390491277120

34438375518437082118207222225979167289125630480117753

N4 =424776472040588014999968946204385846953020717374321677320882

09568215006571676202953757168308992222942860506726420

N5 =101621039523768120659840683670342869972130740083294414947787

21098448355310617860988252607807756465444631925346864 verify {d1, d2, d3} ∈ ⋂_{i=1}^{5} D(N_{i}). Acknowledgment.

I wish to thank Ken Ono for many helpful comments and suggestions in the preparation of this paper, and the referee for his comments for the improvement to the final presentation of the paper.

References

1. P. Erdős and M. Rosenfeld, The factor-difference set of integers, Acta Arith. LXXIX.4 (1997), 353–359.

2. J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin, New York and Heidelberg, 1985. e-mail: jorge.jimenez@uam.es