The number of Diophantine quintuples

Abstract

Abstract. A set {a 1 , . . . , am} of m distinct positive integers is called a Diophantine m-tuple if a i a j + 1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a, b, c} with a < b < c, the number of Diophantine quintuples {a, b, c, d, e} with c < d < e is at most four. Using this result, we further show that the number of Diophantine quintuples is less than 10 276 , which improves the bound 10 1930 due to Dujella.

Any Diophantine quintuple contains a regular Diophantine quadruple

“…Section 3) with Baker's theory (a theorem [14] of Matveev) yields d > b 5 whenever b > 10 50 , which contradicts c = a + b + 2r and d = d + (cf. Corollary 4.5 in [11]). The problem is that there are so many pairs {a, b} with b 10 50 that we cannot apply the reduction method (cf.…”

“…However, if {a, b, c, d, e} is a Diophantine quintuple with c = a + b + 2r < d = d + < e, then {a, c, d} is a "standard triple" (cf. Definition 1 in [8] or Definition 3.1 in [11]). Hence, combining the congruence method (cf.…”

The D(-k2)-triple {1,k2+1,k2+4} with k prime

“…On the other hand, Dujella ([5]) proved that if n ≡ 2 (mod 4), and if n / ∈ S = {−4, −3, −1, 3, 5, 8, 12, 20}, then there exists at least one D(n)-quadruple, and he conjectured that there does not exist a D(n)-quadruple for n ∈ S. Recently, there are numerous papers on this subject, specially in the cases n = 1, n = −1 and n = 4. In particular, Dujella ([8]) proved that there does not exist a D(1)-sextuple and the second author ( [14]) proved that there are at most 10 276 D(1)-quintuples. For the full list of references the reader can see http://web.math.hr/∼duje/dtuples.html.…”

Bounds for Diophantine quintuples

“…In [8] Dujella obtained results very close to settling this conjecture by proving that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. Later bounds on the number of Diophantine quintuples are provided in [9,11,16]. Historical and recent developments of the study of Diophantine m-tuples are found on Dujella's webpage: http://web.math.pmf.unizg.hr/∼duje/dtuples.html.…”