A positive integer is a good number, if its base 10 representation can be split into at least 5 sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has 1 or 2 numbers, it is also a geometric sequence), for example 20240327 is a good number, since after splitting it as 2∣02∣403∣2∣7, 2∣02∣2 and 403∣7 form two groups of geometric sequences.If a>1, m>2, p=1+a+a2+⋯+am is a prime, prove that p10p−1−1 is a good number. number theoryDigitsgeometric sequence