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Diophantus Property

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A set of m distinct positive integers S={a_1,...,a_m} satisfies the Diophantus property D(n) of order n (a positive integer) if, for all i,j=1, ..., m with i!=j,

a_ia_j+n=b_(ij)^2,
(1)

the b_(ij)s are integers. The set S is called a Diophantine n-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: {1,3,8,120} (Davenport and Baker 1969, Jones 1976). There are no others with largest term <=200, and Davenport and Baker (1969) showed that if c+1, 3c+1, and 8c+1 are all squares, then c=120.

General D(1) quadruples are

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

where F_n are Fibonacci numbers, and

{n,n+2,4n+4,4(n+1)(2n+1)(2n+3)}.
(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 D(F_n^2) (Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples D(4k+2).

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={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 c_(-1)(x)=c_0(x)=0, then the general c_k(x) is given by the recurrence relation

c_k=(4x^2+8x+2)c_(k-1)-c_(k-2)+4(x+1).
(5)

The first few c_k are

c_1=4(1+x)
(6)
c_2=4(3+11x+12x^2+4x^3)
(7)
c_3=8(3+23x+62x^2+74x^3+40x^4+8x^5).
(8)

Letting x=1 gives the sequence s_n=1, 3, 8, 120, 1680, 23408, 326040, ... (OEIS A051047), for which sqrt(s_ns_(n+1)+1) is 2, 5, 31, 449, 6271, 87361, ... (OEIS A051048).

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References

Brown, E. "Sets in Which xy+k is Always a Square." Math. Comput. 45, 613-620, 1985.Davenport, H. and Baker, A. "The Equations 3x^2-2=y^2 and 8x^2-7=z^2." 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 m-Tuples-Introduction." http://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 y^2-3x^2=-2 and z^2-8x^2=-7." 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 N+1=x^2, 3N+1=y^2, 8N+1=z^2." 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.

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Diophantus Property

Cite this as:

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

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