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diophantine p_1 p_2...p_8 (x_1/p_1 + ... +x_8/p_8)=N

Source: 2021 Mediterranean Mathematical Olympiad P2 MMC

September 11, 2021
number theoryDiophantine equationdiophantine

Problem Statement

For every sequence p1<p2<⋯<p8p_1<p_2<\cdots<p_8 of eight prime numbers, determine the largest integer NN for which the following equation has no solution in positive integers x1,…,x8x_1,\ldots,x_8: p1 p2 ⋯ p8(x1p1+x2p2+ ⋯ +x8p8)  =  Np_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N
Proposed by Gerhard Woeginger, Austria
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