Diophantine m tuples for primes

Diophantine m-tuples for primes
Andrej Dujella and Florian Luca
Abstract
In this paper, we show that if pis a prime and if A={a1, a2, . . . , am}
is a set of positive integers with the property that aiaj+pis a perfect
square for all 1 ≤i < j ≤m, then m < 3·2168. More generally, when
pis replaced by a squarefree integer n, the inequality m≤f(ω(n))
holds with some function f, where ω(n) is the number of prime divi-
sors of n. We also give upper bounds for mwhen pis replaced by an
arbitrary integer which hold on a set of nof asymptotic density one.
1 Introduction
Let nbe any nonzero integer. A Diophantine m-tuple with the property D(n)
is a set of mpositive integers {a1, . . . , am}such that aiaj+nis a perfect square
for all 1 ≤i < j ≤m. Diophantus found the quadruple {1,33,68,105}
with the property D(256), while the first Diophantine quadruple with the
property D(1), the set {1,3,8,120}, was found by Fermat (see [4]). Baker
and Davenport [1] proved that this Fermat set cannot be extended to a
Diophantine quintuple. The first author proved recently that there does not
exist a Diophantine sextuple with the property D(1), and that there are only
finitely many such quintuples (see [7]). On the other hand, there are examples
of Diophantine sextuples, e.g. {99,315,9920,32768,44460,19534284}with
the property D(2985984), found by Gibbs [9].
The question is what can be said about the size of sets with the property
D(n) for n6= 1. Let
Mn= sup{|S| :Shas the property D(n)}.
02000 Mathematics Subject Classification 11D45.
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Considering congruences modulo 4, it is easy to see that Mn= 3 if n≡2
(mod 4) (see [2]). On the other hand, if nis not congruent to 2 modulo 4
and n6∈ {−4,−3,−1,3,5,8,12,20}, then Mn≥4 (see [5]).
Since the number of integer points on the elliptic curve
y2= (a1x+n)(a2x+n)(a3x+n)
is finite, we conclude that there does not exist an infinite set with the property
D(n). Furthermore, the hyperelliptic curve
y2= (a1x+n)(a2x+n)(a3x+n)(a4x+n)(a5x+n)
has genus g= 2. Caporaso, Harris and Mazur [3], proved that the Lang
conjecture on varieties of general type implies that for g≥2 the number
B(g, K) = maxC|C(K)|is finite. Here, Cruns over all curves of genus g
over a number field K, and C(K) denotes the set of all K-rational points on
C. However, even the question whether B(2,Q)<∞is still open. Since
Mn≤5 + B(2,Q) (by [11], we also have Mn≤4 + B(4,Q)), we see that the
Lang conjecture implies that there exist an absolute constant Csuch that
Mn≤Cfor all nonzero integers n. However, at present, the best known
upper bound for Mnhas the form Mn≤Clog |n|(see [6, 8]).
The main result of this paper consists in an absolute upper bound for the
size of sets with properties D(p) and D(−p), where pis a prime.
Throughout the paper, the letter pwill always denote a prime number.
For a nonzero integer n, we write ω(n) and P(n) for the number of prime
divisors and the largest prime factor of n, respectively, with the convention
that P(±1) = 1. As usual, π(x) denotes the number of primes p≤x. We
use the Vinogradov symbols ≪and ≫, as well as the Landau symbols Oand
o, with their usual meanings.
Acknowledgments. The authors would like to thank the referee for
valuable comments. This paper was written during a visit of the second
author at the University of Zagreb in October of 2004. He warmly thanks
this University for its hospitality. Both authors were partly supported by the
Croatian Ministry of Science, Education and Sport Grant 0037110.
2 Results
The first result of this paper is an absolute upper bound on the size mof a
Diophantine m-uple with the property D(±p) which holds for all primes p.
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Theorem 2.1. There exists an absolute constant Csuch that any Diophan-
tine m-tuple with the property D(p)or D(−p), where pis a prime, satisfies
m < C. Furthermore, Ccan be chosen to be 3·2168.
We next give a more general result, namely an upper bound on the size
mof a Diophantine m-tuple with the property D(±n), where nis squarefree,
which depends only on the number of prime divisors ω(n) of the positive
integer n.
Theorem 2.2. There exists a function f: IN −→ IN such that if nis any
squarefree positive integer, then any Diophantine m-tuple with the property
D(n)or D(−n)satisfies m < f(ω(n)), where ω(n)is the number of distinct
prime divisors of n.
We finally present an upper bound on mwhich is valid for most positive
integers n. Let us call a Diophantine m-tuple Awith the property D(n)
reduced if gcd(a, n) = 1 holds for all a∈ A.
Theorem 2.3. For every ε > 0, the set of positive integers nwith the prop-
erty that there exists a Diophantine m-tuple with the property D(n)or D(−n)
and with m > (1 + ε) log log n, is of asymptotic density zero. Furthermore,
if only reduced Diophantine m-tuples are considered, then the same result
holds with (1 + ε) log log nreplaced by any increasing function f(n)such that
limx→∞ f(x) = +∞.
3 The proof of Theorem 2.1
We shall analyze in detail the case of the Diophantine tuples with the prop-
erty D(p), and we shall only point out the minor differences in the argument
for the case of the Diophantine tuples with the property D(−p).
We start with a short outline of the methods used in the proof. In Section
3.1 we show that in order to prove Theorem 2.1, it suffices to establish a gap
principle of the form ai+ℓ> pγaifor the elements of a Diophantine tuple.
In the first step, in Section 3.2, a factorization is assigned to any triple
with the property D(p). A combination of p-adic and archimedean estimates
suffice to determine that the second factor appearing in the factorization
should be divisible by p2, since divisibility of the first factor leads to a quick
conclusion of the proof.
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In the second step, we obtain in Lemma 3.1 a congruence condition as-
sociated with solutions and deduce from this a set of new polynomial con-
gruences. By elimination of variables, one gets a new polynomial of bounded
height which must satisfy a congruence modulo pfor rather small values of
the variables if the gap principle we want is not fulfilled. The smallness of
the variables yields the key fact that the congruence in question must be in
fact an equation.
The final step consists in deducing from this the existence of a polynomial
of bounded degree Min five variables which vanishes on a set which is a
cartesian product M5, with Ma set of cardinality larger than M, which is
impossible. This establishes the gap principle.
3.1 Reductions of the original problem
Let A={a1, . . . , am}be a Diophantine m-tuple with the property D(p).
Since the main result from [8] shows that if Ais any Diophantine m-tuple
with the property D(n) (nany nonzero integer), then m≤16 log |n|for
|n|>400, and m≤31 for |n| ≤ 400, it follows that from now on we may
assume that p > 2276 , and that C > 280 log 2.
We assume that a1< a2<··· < am. Furthermore, whenever we write
aiaj+p=x2
ij, we make the convention that xij >0. We note that there
exists at most one element of Awhich is a multiple of p. Indeed, for if two
such elements, say aiand ajexist, then reducing the equation aiaj+p=x2
ij
modulo p2we get p≡x2
ij (mod p2), which is a contradiction. Eliminating
such an element from A, it follows that we may assume that pdoes not divide
afor any a∈ A. We now note that a3> p1/4. Indeed, note that if we write
a1a3+p=x2
13 and a2a3+p=x2 23,
then x23 > x13 > p1/2. In particular,
a2
3> a3(a2−a1) = x2 23 −x2 13 = (x23 −x13)(x23 +x13)≥2p1/2.
Hence, a3> p1/4. When pis replaced by −p, we then have
a2
2> a1a2=p+x2 12 > p,
therefore a2> p1/2, which is even a better inequality. Thus, eliminating the
smallest two elements, if needed, we may assume that a > p1/4holds for all
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a∈ A. Furthermore, a result from [6] shows that there exist at most 21
elements a∈ A such that a > p3. Eliminating those elements too, we may
assume that p1/4< a < p3holds for all a∈ A. We now note that if there
exist constants γand ℓsuch that the inequality ai+ℓ> pγaiholds for all
i∈ {1, . . . , m −ℓ}, then, by induction on i, the inequality
ai> p1/4+⌊i/ℓ⌋γ
holds for i= 1, . . . , m. Since am< p3, we get the inequality
jm
ℓkγ+1 4<3.
Hence,
m < (11 + 4γ)ℓ
4γ.(1)
The above argument together with the fact that we first eliminated at most
24 elements from A, shows that one may take
C= max 280 log 2,(11 + 4γ)ℓ
4γ+ 24.(2)
Thus, it suffices to find such constants γand ℓ.
3.2 A factorization
To any triple with the property D(p), a factorization can be assigned. We will
show that p2divides exactly one of the factors appearing in the factorization.
The case when the smaller factor is divisible by p2leads easily to desired
inequality of the form ai+ℓ> pγai. The case when the larger factor is divisible
by p2is much more involved, and will be considered later.
Let a < b < c be any three elements in A. We write
ab +p=x2, bc +p=y2and ac +p=z2,
where x,yand zare positive integers. Then,
(xyz)2= (ab +p)(bc +p)(ac +p)
= (abc)2+pabc(a+b+c) + p2(ab +bc +ac) + p3
= (abc +p(a+b+c)/2)2
+p2ab +ac +bc −(a+b+c)2/4+p3,
5