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Understanding Math Beyond the Classroom
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How it Works › Forums › Exploration Problems (Interesting Math Problems!) › Foundations of NT Exploration
Here are some exploration problems (food for thought) that will lead you to find some of the most important mathematical discoveries of this era! Here are 3 exploration problems relating to the Foundations of NT. As always, each question adds 10 points to your leaderboard score!
1. Investigate $x^6 \bmod{7}$ for $x \equiv 1,2,3,4,5,6$. What patterns do you notice? In general, what do you notice for $x^{p-1} \bmod{p}$ for $x \equiv 1,2,3,…,p-1$? Going even further, what do you notice of $x^{\phi(n)} \bmod{n}$ for x and n such that $gcd(x,n) = 1$?\\
2. Try and find a worded proof for why the formula for $\phi(n)$ might make sense. Can you prove it mathematically?
3. Investigate $(n-1)! \bmod{n}$ when $n$ is prime. Can you find a formula? Can you prove it? What about when n is composite?
If you have any questions, feel free to post in the Questions forum.