Diophantus Property -- from Wolfram MathWorld

Diophantus Property

A set of distinct positive integers $S={a_1,...,a_m}$ 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), ... (Sloane's A050269 and A050270). Diophantine 1-triples are less abundant: (1, 3, 8), (2, 4, 12), (1, 8, 15), (3, 5, 16), (4, 6, 20), ... (Sloane's 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

${F_(2n),F_(2n+2),F_(2n+4),4F_(2n+1)F_(2n+2)F_(2n+3),}$

(2)

where are Fibonacci numbers, and

(3)

The quadruplet

${2F_(n-1),2F_(n+1),2F_n^3F_(n+1)F_(n+2),2F_(n+1)F_(n+2)F_(n+3)(2F_(n+1)^2-F_n^2)}$

(4)

is (Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples .

Jones (1976) derived an infinite sequence of polynomials $S={x,x+2,c_1(x),c_2(x),...}$ 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, ... (Sloane's A051047), for which is 2, 5, 31, 449, 6271, 87361, ... (Sloane's 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.

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.

Morgado, J. "Generalization of a Result of Hoggatt and Bergum on Fibonacci Numbers." Portugaliae Math. 42, 441-445, 1983-1984.

Sloane, N. J. A. Sequences A050269, A050269, A050273, A050274, A050275, A051047, and A051048 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Diophantus Property." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantusProperty.html