Diophantus Property
A set of 
distinct positive integers ![S=<span style=]()
{a_1,...,a_m}" src="https://mathworld.wolfram.com/images/equations/DiophantusProperty/Inline2.gif" style="height:15px; width:93px" /> satisfies the Diophantus property 
of order 
(a positive integer) 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: ![<span style=]()
{1,3,8,120}" src="https://mathworld.wolfram.com/images/equations/DiophantusProperty/Inline11.gif" style="height:15px; width:76px" /> (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
![<span style=]() {F_(2n),F_(2n+2),F_(2n+4),4F_(2n+1)F_(2n+2)F_(2n+3),} " src="https://mathworld.wolfram.com/images/equations/DiophantusProperty/NumberedEquation2.gif" style="height:15px; width:241px" /> |
(2)
|
where 
are Fibonacci numbers, and
![<span style=]() {n,n+2,4n+4,4(n+1)(2n+1)(2n+3)}. " src="https://mathworld.wolfram.com/images/equations/DiophantusProperty/NumberedEquation3.gif" style="height:15px; width:260px" /> |
(3)
|
The quadruplet
![<span style=]() {2F_(n-1),2F_(n+1),2F_n^3F_(n+1)F_(n+2),2F_(n+1)F_(n+2)F_(n+3)(2F_(n+1)^2-F_n^2)} " src="https://mathworld.wolfram.com/images/equations/DiophantusProperty/NumberedEquation4.gif" style="height:21px; width:368px" /> |
(4)
|
is 
(Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples 
.
A longstanding conjecture is that no integer Diophantine quintuple exists (Gardner 1967, van Lint 1968, Davenport and Baker 1969, Kanagasabapathy and Ponnudurai 1975, Sansone 1976, Grinstead 1978).
Jones (1976) derived an infinite sequence of polynomials ![S=<span style=]()
{x,x+2,c_1(x),c_2(x),...}" src="https://mathworld.wolfram.com/images/equations/DiophantusProperty/Inline21.gif" style="height:15px; width:177px" /> 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
Letting 
gives the sequence 
, 3, 8, 120, 1680, 23408, 326040, ... (OEIS A051047), for which 
is 2, 5, 31, 449, 6271, 87361, ... (OEIS A051048).
REFERENCES:
Brown, E. "Sets in Which 
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: 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." https://web.math.hr/~duje/intro.html.
Gardner, M. "Mathematical Diversions." Sci. Amer. 216, 124, 1967.
Grinstead, C. M. "On a Method of Solving a Class of Diophantine Equations." Math. Comput. 32, 936-940, 1978.
Hoggatt, V. E. Jr. and Bergum, G. E. "A Problem of Fermat and the Fibonacci Sequence." Fib. Quart. 15, 323-330, 1977.
Jones, B. W. "A Variation of a Problem of Davenport and Diophantus." Quart. J. Math. (Oxford) Ser. (2) 27, 349-353, 1976.
Kanagasabapathy, P. and Ponnudurai, T. "The Simultaneous Diophantine Equations 
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 A050269, A050269, A050273, A050274, A050275, A051047, and A051048 in "The On-Line Encyclopedia of Integer Sequences."
van Lint, J. H. "On a Set of Diophantine Equations." T. H.-Report 68-WSK-03. Department of Mathematics. Eindhoven, Netherlands: Technological University Eindhoven, 1968.
Referenced on Wolfram|Alpha: Diophantus Property
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