MathDB

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

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+a^2+\dots+a^m

is a prime, prove that

\frac{10^{p-1}-1}{p}

is a good number.

Problem Statement