Diophantine quintuple

☆ Save On Wikipedia ↗

In number theory, 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}\}} {\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} {\displaystyle a_{i}a_{j}+1} is a perfect square for any 1 ≤ i < j ≤ m . {\displaystyle 1\leq i<j\leq m.} {\displaystyle 1\leq i<j\leq m.}[1] 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.

Diophantine m-tuples

The first diophantine quadruple was found by Fermat: { 1 , 3 , 8 , 120 } . {\displaystyle \{1,3,8,120\}.} {\displaystyle \{1,3,8,120\}.}[1] It was proved in 1969 by Baker and Davenport [1] 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 {\tfrac {777480}{8288641}}.} {\displaystyle {\tfrac {777480}{8288641}}.}[1]

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.[1] In 2016 it was shown that no such quintuples exist by He, Togbé and Ziegler.[2]

As Euler proved, every Diophantine pair can be extended to a Diophantine quadruple. The same is true for every Diophantine triple. In both of these types of extension, as for Fermat's quadruple, it is possible to add a fifth rational number rather than an integer.[3]

The rational case

Diophantus himself found the rational diophantine quadruple { 1 16 , 33 16 , 17 4 , 105 16 } . {\displaystyle \left\{{\tfrac {1}{16}},{\tfrac {33}{16}},{\tfrac {17}{4}},{\tfrac {105}{16}}\right\}.} {\displaystyle \left\{{\tfrac {1}{16}},{\tfrac {33}{16}},{\tfrac {17}{4}},{\tfrac {105}{16}}\right\}.}[1] More recently, Philip Gibbs found sets of six positive rationals with the property.[4] 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.[5]

References

  1. Dujella, Andrej (January 2006). "There are only finitely many Diophantine quintuples". Journal für die reine und angewandte Mathematik. 2004 (566): 183–214. CiteSeerX 10.1.1.58.8571. doi:10.1515/crll.2004.003.
  2. He, B.; Togbé, A.; Ziegler, V. (2016). "There is no Diophantine Quintuple". Transactions of the American Mathematical Society. arXiv:1610.04020.
  3. Arkin, Joseph; Hoggatt, V. E. Jr.; Straus, E. G. (1979). "On Euler's solution of a problem of Diophantus" (PDF). Fibonacci Quarterly. 17 (4): 333–339. doi:10.1080/00150517.1979.12430206. MR 0550175.
  4. Gibbs, Philip (1999). "A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples". arXiv:math.NT/9903035v1.
  5. Herrmann, E.; Pethoe, A.; Zimmer, H. G. (1999). "On Fermat's quadruple equations". Math. Sem. Univ. Hamburg. 69: 283–291. doi:10.1007/bf02940880. hdl:2437/90714.