Math 453: (Fake) Homework 14

(Fake) Homework 14

This is a fake homework assignment, which means that you do not have to complete this assignment nor turn it in. It is provided here so you'll have an opportunity to practice some of the ideas discussed in the most recent lectures.

Although the harmonic series diverges, it is an old theorem of Euler that the series $\sum_{k=1}^n \frac{1}{k}-\ln(n)$ does converge. The limit is known as Euler's constant, also written as $\gamma = \lim_{n \to \infty} \left(\sum_{k=1}^n \frac{1}{k}-\ln(n)\right)$ , and it is a very important number. The decimal approximation for $\gamma$ is $0.577215664901532860606512090082402431042$ . Use this to compute the first several terms of the continued fraction expansion of $\gamma$ , and then find the best rational approximation to $\gamma$ with denominator less than 20.
Find the first number greater than 1000 which cannot be expressed as a sum of three cubes.
Determine whether $1729$ and $6409$ can be written as sums of squares. If so, do it; if not, give a complete justification.
Composite numbers X and Y are chosen, and the product is given to Megan and the sum is given to Anthony. If Megan says she cannot compute Anthony's sum, explain why you know that her product is not $625$ . If Anthony claims that he knew she couldn't have known the sum, explain why his sum cannot be $36$ .