December PoTM

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Written by Eileen Wu

 

Submit your answers in the Google Form: https://forms.gle/gUfEW1JWKT1bqBDs8.

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Stuck? Check out these resources and hints:

 

Resources: 

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AoPS page on modular arithmetic

Hints:

 

1. Compute a few small terms. Do you notice anything about these values?

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2. Expand (6+√35)^2. Try to rewrite it using only integers and (6+√35). Do the same for (6-√35)^2. See if you notice a pattern.

 

3. If (6+√35)^2 can be written using (6+√35) and a constant, then (6+√35)^n can be written using earlier powers. The same applies to (6-√35)^n. This means any term can be expressed in terms of the previous two terms.

 

4. Once you find a recurrence, reduce it modulo 7. Compute several terms mod 7 and look for a repeating pattern.