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ni of pos. integer solutions of x_1 + 2x_2 + 3x_3 + ... + nx_n = a

Source: Eotvos 1904 p2

September 6, 2024
number theorydiophantine

Problem Statement

If a is a natural number, show that the number of positive integral solutions of the indeterminate equation x1+2x2+3x3+...+nxn=a  (1)x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1) is equal to the number of non-negative integral solutions of y1+2y2+3y3+...+nyn=a−n(n+1)2  (2)y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2) [By a solution of equation (1), we mean a set of numbers {x1,x2,...,xn}\{x_1, x_2,..., x_n\} which satisfies equation (1)].
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