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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.

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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?

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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

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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.

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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.

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