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diophantine system nx(n-1) has a solution, therefor n is even

Source: 1983 Swedish Mathematical Competition p3

March 28, 2021
Evennumber theorysystem of equationsSystemdiophantineDiophantine equation

Problem Statement

The systems of equations {2x1−x2=1−x1+2x2−x3=1−x2+2x3−x4=1−x3+3x4−x5=1⋯⋯⋯⋯−xn−2+2xn−1−xn=1−xn−1+2xn=1\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. has a solution in positive integers xix_i. Show that nn must be even.
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