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## 13S Minimum prime divisors

13. [PEN A14 A71]

A14. Let $n>1$ be an integer. Show that $n$ does not divide $2^n-1$.

A71. Determine all integers $n>1$ such that $\frac{2^n +1}{n^2}$ is an integer.
Here is the official solution file: pen13.pdf

You can also discuss the problems: A14 & A71

## 13 Minimum prime divisors

13. [PEN A14 A71]

A14. Let $n>1$ be an integer. Show that $n$ does not divide $2^n-1$.

A71. Determine all integers $n>1$ such that $\frac{2^n +1}{n^2}$ is an integer.

## 12S A Generalization of an Identity

12. PEN I10 Show that for all primes $p$,

$\sum_{k=1}^{p-1}{\left\lfloor\frac{k^{3}}{p}\right\rfloor}=\frac{(p+1)(p-1)(p-2)}{4}$.

Here is the official solution file: pen12.pdf
You can also discuss the problems here!

12. PEN I10 Show that for all primes $p$,
$\sum_{k=1}^{p-1}{\left\lfloor\frac{k^{3}}{p}\right\rfloor}=\frac{(p+1)(p-1)(p-2)}{4}$.