## Generalization of a problem of Diophantus.(English)Zbl 0849.11018

Let $$n$$ be an integer. The set of natural numbers $$\{a_1, a_2, \dots, a_m\}$$ has the property of Diophantus of order $$n$$, in brief $$D(n)$$, if $$i\neq j$$ for all $$i,j= 1, 2,\dots, m$$ and the following holds: $$a_i a_j+ n= b^2_{ij}$$, where $$b_{ij}$$ is an integer. Some problems of the existence of sets of four natural numbers with property $$D(n)$$, for any integer $$n$$, are considered. The main results are: (1) If $$n$$ is an integer of the form $$n= 4k+2$$ ($$k$$ an integer), then there does not exist a set of four natural numbers with the property $$D(n)$$; (2) If $$n$$ is an integer not of the form $$4k+2$$, and $$n\not\in S=\{ 3, 5, 8, 12, 20, -1, -3, -4\}$$, then there exists at least one set of four natural numbers with the property $$D(n)$$.
Reviewer: E.L.Cohen (Ottawa)

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11D09 Quadratic and bilinear Diophantine equations

### Keywords:

perfect squares; Fibonacci numbers; property of Diophantus
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