Takehome Midterm

Due Date: Friday, March 18 at 11:59 pm

Rules: This is a midterm exam, not a homework assignment. You must solve the problems entirely on your own, and you should not discuss the problems with any other students in the class. You should feel free to read internet web pages that might be helpful (e.g. Wikipedia articles and online documentation for Mathematica and SageMath), but you should not solicit help by posting questions to any online forums.

Note: You should feel free to use any of the built-in number theory commands in either Mathematica or SageMath. Click on the following links to get a list of potentially useful commands:

• Number Theory in Mathematica
• Number Theory in SageMath

1. Find a positive integer \(n\) so that \(n^3 + n \equiv 3\;(\mathrm{mod}\;8141081016796875)\).
2. Prove that if \(p\) is prime then \( p^6 + 14p + 20\) is composite.
3. Let \(a_n = n^2 + 1024\) for all \(n\in\mathbb{N}\). What is the maximum value of \(\mathrm{gcd}(a_n, a_{n+1})\)? Prove your answer.
4. If \(n\) is a positive integer, let \(d(n)\) denote the number of positive divisors of \(n\), including both \(1\) and \(n\). Find a positive integer \(n\) so that \(d(n) \geq 1000\) and \(d(n+1) \geq 1000\).
5. Find a positive integer \(n\) for which \(\phi(n) < n/7\), where \(\phi\) denotes Euler's totient function.