2

Part of 2021 Mediterranean Mathematics Olympiad

Problems(1)

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

Source: 2021 Mediterranean Mathematical Olympiad P2 MMC

For every sequence

p_1<p_2<\cdots<p_8

of eight prime numbers, determine the largest integer

N

for which the following equation has no solution in positive integers

x_1,\ldots,x_8

p_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

number theoryDiophantine equationdiophantine