The Diophantus property 
of order 
(a positive integer) is
satisfied by a set 
of 
distinct positive integers if, for all 
, ..., 
with 
,
 |
(1)
|
the 
s
are integers. The set 
is called a Diophantine 
-tuple.
Diophantine 1-doubles are abundant: (1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (1, 8), (3, 8), (6, 8), (7, 9), (8, 10), (9, 11), ... (OEIS A050269 and A050270). Diophantine 1-triples are less abundant: (1, 3, 8), (2, 4, 12), (1, 8, 15), (3, 5, 16), (4, 6, 20), ... (OEIS A050273, A050274, and A050275).
Fermat found the smallest Diophantine 1-quadruple: 
(Davenport and Baker 1969, Jones 1976). There are
no others with largest term 
, and Davenport and Baker (1969) showed that if 
, 
, and 
are all squares, then 
.
General 
quadruples are
 |
(2)
|
where 
are Fibonacci numbers, and
 |
(3)
|
The quadruplet
 |
(4)
|
is 
(Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples 
.
The longstanding conjecture that no 
quintuple of distinct positive
integers exists (Gardner 1967, van Lint 1968, Davenport and Baker 1969, Kanagasabapathy
and Ponnudurai 1975, Sansone 1976, Grinstead 1978) was proved by He et al. (2019).
Jones (1976) derived an infinite sequence of polynomials 
such that the product of any two
consecutive polynomials, increased by 1, is the square of a polynomial. Letting 
, then the general
is given by the recurrence relation
 |
(5)
|
The first few 
are
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
Letting 
gives the sequence 
,
3, 8, 120, 1680, 23408, 326040, ... (OEIS A051047),
for which 
is 2, 5, 31, 449, 6271, 87361, ... (OEIS A051048).
Is Always a Square." Math. Comput. 45,
613-620, 1985.Davenport, H. and Baker, A. "The Equations 
and 
." Quart. J. Math. (Oxford) Ser. 2 20,
129-137, 1969.Diofant Aleksandriĭskiĭ. Arifmetika i kniga
o mnogougol'nyh chislakh [Russian]. Moscow, Russia: Nauka, 1974.Dujella,
A. "Generalization of a Problem of Diophantus." Acta Arith. 65,
15-27, 1993.Dujella, A. "Diophantine Quadruples for Squares of
Fibonacci and Lucas Numbers." Portugaliae Math. 52, 305-318, 1995.Dujella,
A. "Generalized Fibonacci Numbers and the Problem of Diophantus." Fib.
Quart. 34, 164-175, 1996.Dujella, A. "Diophantine 
-Tuples-Introduction." 
and 
." Quart. J. Math. (Oxford) Ser. (2) 26,
275-278, 1975.Morgado, J. "Generalization of a Result of Hoggatt
and Bergum on Fibonacci Numbers." Portugaliae Math. 42, 441-445,
1983-1984.Sansone, G. "Il sistema diofanteo 
, 
, 
." Ann. Mat. Pura Appl. 111, 125-151,
1976.Sloane, N. J. A. Sequences