Some rational Diophantine sextuples

Abstract

Abstract. A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm used to find them is described. A classification of quadruples and quintuples with examples and statistics is also given.

Fibonacci Diophantine triples

“…This was almost proved by Dujella [5], who showed that there can be at most finitely many Diophantine quintuples and all of them are, at least in theory, effectively computable. In the rational case, it is not known that the size m of the Diophantine m-tuples must be bounded and a few examples with m = 6 are known by the work of Gibbs [8]. We also note that some generalization of this problem for squares replaced by higher powers (of fixed, or variable exponents) were treated by many authors (see [1,2,9,13] and [10]).…”

Lucas Diophantine Triples

“…In the rational case, it is not known if the size m of the Diophantine m-tuples must be universally bounded. A few examples with m = 6 are known by the work of Gibbs [7]. Several generalizations of this problem, when the squares are replaced by higher powers of fixed, or variable exponents, were treated in many papers (see [1], [2], [8], [9]) and [10]).…”

On the extensibility of Diophantine triples {k-1, k+1, 4k} for Gaussian integers

“…. , a m } in a commutative ring R with 1 is called Diophantine m-tuple if a i a j + 1 is a perfect square in R for all 1 ≤ i < j ≤ m. Let us mention the most famous historical examples of such sets: the first rational quadruple { 16 } found by Gibbs ( [13]). There exist families of such sets, for instance quadruples {F 2k , F 2k+2 , F 2k+4 , 4F 2k+1 F 2k+2 F 2k+3 } (where F k is k-th Fibonacci number) and {k − 1, k + 1, 4k, 16k 3 − 4k} (which both represent a generalization of the Fermat's quadruple).…”