pdf: included in Caderno (see “caderno” page)
I submitted a solution to the following Problem that was accepted.
Find the length of the period of the repeating decimal representation of
The repeating decimal representation of the number is
Let be a prime number. The period of the repeating decimal of is equal to the order of and is either or a divisor of Since is a prime number, the period of the repeating decimal of is therefore either or a divisor of . These divisors are
By the definition of the order of , I have to find the smallest of these such that
which means should be an integer.
the remaining cases are . From these I have checked in PARI that only
is an integer (*). For instance
Conclusion. The length of the period of the repeating decimal representation of is
(*) Edited a little bit to improve the English text.
A much more sophisticated and elegant solution by Philipp Lampe was posted by one of the authors of this excellent advanced mathematical blog.