Diophantine quintuple

In mathematics, a diophantine m-tuple is a set of m positive integers { a 1 , a 2 , a 3 , a 4 , … , a m } {displaystyle {a_{1},a_{2},a_{3},a_{4},ldots ,a_{m}}} such that a i a j + 1 {displaystyle a_{i}a_{j}+1} is a perfect square for any 1 ≤ i < j ≤ m {displaystyle 1leq i<jleq 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. In mathematics, a diophantine m-tuple is a set of m positive integers { a 1 , a 2 , a 3 , a 4 , … , a m } {displaystyle {a_{1},a_{2},a_{3},a_{4},ldots ,a_{m}}} such that a i a j + 1 {displaystyle a_{i}a_{j}+1} is a perfect square for any 1 ≤ i < j ≤ m {displaystyle 1leq i<jleq 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 } {displaystyle {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 {displaystyle {frac {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 it was shown that no such quintuples exist by He, Togbé and Ziegler. Diophantus himself found the rational diophantine quadruple { 1 16 , 33 16 , 17 4 , 105 16 } {displaystyle left{{frac {1}{16}},{frac {33}{16}},{frac {17}{4}},{frac {105}{16}} ight}} . 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.