In mathematics, a diophantine m-tuple is a set of m positive integers { a 1 , a 2 , a 3 , a 4 , … , a m } such that a i a j + 1 is a perfect square for any 1 ≤ i < j ≤ m . A set of m positive rational numbers with the similar property that the product of any two is one less than a rational square is known as a rational diophantine m-tuple.
The first diophantine quadruple was found by Fermat: { 1 , 3 , 8 , 120 } . It was proved in 1969 by Baker and Davenport that a fifth positive integer cannot be added to this set. However, Euler was able to extend this set by adding the rational number 777480 8288641 .
The question of existence of (integer) diophantine quintuples was one of the oldest outstanding unsolved problems in Number Theory. In 2004 Andrej Dujella showed that at most a finite number of diophantine quintuples exist. In 2016 the problem was finally resolved by He, Togbé and Ziegler.
Diophantus himself found the rational diophantine quadruple { 1 16 , 33 16 , 17 4 , 105 16 } . More recently, Philip Gibbs found sets of six positive rationals with the property. It is not known whether any larger rational diophantine m-tuples exist or even if there is an upper bound, but it is known that no infinite set of rationals with the property exists.
