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On certain Diophantine equations. (Slovak. English summary) Zbl 0195.33002
A method to find all solutions of the equation
\[ a_0+a_1x_1^2+\ldots+ a_nx_1^2x_2^2\cdots x_n^2 = x_1x_2\cdots x_nx_{n+1} \]
is given; the number of solutions is finite. Define \(\beta_i\) by the formulae \(\beta_1= a_0\), \(\beta_{k+1} = 1 + a_k\beta_1 \beta_2\cdots \)\beta_k\(. The numbers \)\beta_1\(, \)\beta_2, \ldots, \beta_{n+1}\( satisfy the above equation and the numbers \)\beta_2\(, \)\beta_3, \ldots, \(\beta_{n+1}\) satisfy the equation \(a_1/x_1+ a_2/x_2 +\ldots+ a_n/x_n -1 = 1/a_0.\)
MSC:
11D68 Rational numbers as sums of fractions
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References:
[1] Erdős P.: \(Az\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} = \frac{a}{b}\) egyenlet egészszámú megoldásairól. Matematikai Lapok 1 (1950) 192-210.
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