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Poor number with 100 000 distinct prime divisors

Source: Saint Petersburg olympiad 2024, 9.6

September 22, 2024
number theory

Problem Statement

Call a positive integer number nn poor if equation x1x2…x101=(n−x1)(n−x2)…(n−x101)x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) has no solutions in positive integers 1<xi<n1<x_i<n. Does there exist poor number, which has more than 100 000100 \ 000 distinct prime divisors?
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