Yesterday, I challenged you to prove that

where , , and the are defined by and .

The proof is by induction on . The base case is just arithmetic:

Now suppose that we already know the statement holds for some particular ; we must show that it also holds for . The proof is not too hard, but we have to handle the stacked exponents with care! (Note also that all the following equalities are really taken , which is OK since addition, subtraction, and multiplication are all compatible with taking remainders.)

(The last step is because we know from yesterday that .) So , which is what we wanted to show.

## About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

Here it is proved not only the equality mod M_n, but it is also proved the equality in general. Am I right?

Well, we defined the s_n in terms of mod M_n in the first place. I will write more about this today.

Ok. Thanks.

Hendrix College is a University or a high-school? What can we study there, high-school subjects or universitary careers?

The terminology can be confusing since it is used very differently depending on what country you are in. Hendrix is definitely not a high school, it is a 4-year school that a student would attend after completing high school. It is not a university though, the difference being that it is relatively small and does not offer graduate degrees like a master’s degree or PhD.

Ok, thanks for your answer, friend!

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So, here are studied professional careers, but with a duration of 4 years. What careers are studied here, for instance?

I suggest you just go poke around https://www.hendrix.edu/ , I think you will be able to find answers to your questions much more quickly that way than I would be able to answer them. For example there is a list of the academic programs here: https://www.hendrix.edu/academics/majorsandminors/

Thank you!

I had visited that site, but I hadn’t reviewed the academics section.

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