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On the solutions of equation \(\sum\limits^s_{i=1}\dfrac1{x_i}+\dfrac1{x_1\cdots x_s}=1\) and the number of solutions of Znám’s problem. (Chinese. English summary) Zbl 1199.11084

Summary: Let \(s\) be positive integer. Let \(\varOmega (s)\), \(Z(s)\) and \(H(s)\) denote the number of solutions of the equation \(\sum\limits^s_{i=1}\dfrac1{x_i}+\dfrac1{x_1\cdots x_s}=1\), Znám’s problem and the system of congruences \(x_1\cdots x_{i-1}x_{i+1}\cdots x_s+1\equiv 0 (\bmod\;x_i)\), respectively. The authors give two new methods to structure solutions of the equation, and prove that \(\varOmega (8)\geq73,\varOmega (9)\geq 279,\varOmega (10)\geq576\). Meanwhile, the number of solutions of the equation is further improved. It is proved that if \(2|s\geq 12\), then \(\varOmega (s+1)\geq \varOmega (s)+101\) and if \(2\nmid s\geq 11\), then \(\varOmega (s+1)\geq\varOmega (s)+70\).

MSC:

11D68 Rational numbers as sums of fractions
11D45 Counting solutions of Diophantine equations
11D79 Congruences in many variables
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