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## Archive for December, 2008

C2 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Here is the official solution file: pen16.pdf

You can discuss the problem in MathLinks here.

C2 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

## 15. Exponential Congruence Sequence

D5. Prove that for , $\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.$
D6. Show that, for any fixed integer  the sequence $2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}$ is eventually constant.

Sorry all, for the delay of the problem 15.  Here goes the solution: pen-15.pdf

## 14S. Different Approaches to an Intuitive Problem

The fourth problem of the second season of PEN is as follows:

N17. Suppose that $a$ and $b$ are distinct real numbers such that:
$a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots$ are all integers. Show that $a$ and $b$ are integers.

Here is the official solution file: pen14.pdf

You can discuss the problem in MathLinks here.

N17. Suppose that $a$ and $b$ are distinct real numbers such that:
$a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots$ are all integers. Show that $a$ and $b$ are integers.