Diophantus Property

A set of Positive Integers $S=\{a_1, \ldots, a_m\}$ satisfies the Diophantus property $D(n)$ of order $n$ if, for all $i,j=1$, ..., $m$ with $i\not=j$,

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

where $n$ and $b_{ij}$ are Integers. The set $S$ is called a Diophantine $n$-tuple. Fermat found the first $D(1)$ quadruple: $\{1, 3, 8, 120\}$. 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

$$\{2F_{n-1}, 2F_{n+1}, 2{F_n}^3F_{n+1}F_{n+2}, 2F_{n+1}F_{n+2}F_{n+3}(2{F_{n+1}}^2-{F_n}^2)\}$$ (4)

is $D({F_n}^2)$ (Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples $D(4k+2)$.

References

Aleksandriiskii, D. Arifmetika i kniga o mnogougol'nyh chislakh. Moscow: Nauka, 1974.

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.

Dujella, A. ``Generalization of a Problem of Diophantus.'' Acta Arithm. 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.