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Archive for November, 2008

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

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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.

Read Full Post »

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!

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Here goes the second problem of the second season!

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}.

Read Full Post »