Jim Belk University of Glasgow

Homework 4

Due Date: Friday, February 26

    1. Prove that \(\sqrt[\large 3]{2}+\sqrt[\large 3]{4}\) is irrational.
    2. Prove that \(\cos(\pi/9)\) is irrational.
  2. Let \(a\), \(b\), and \(c\) be pairwise relatively prime positive integers.
    1. Prove that there do not exist positive integers \(x\), \(y\), and \(z\) such that \[ abx + acy +bcz \,=\, 2abc. \]
    2. Prove that if \(d > 2abc\) then there exist positive integers \(x\), \(y\), and \(z\) so that \[ abx + acy +bcz \,=\, d. \]
  3. If \(q\) is a rational number and \(0< q < 1\), the factorial digits of \(q\) are the sequence \(d_1,\ldots,d_n\) of integers that satisfy the following conditions:
    • \(0\leq d_k \leq k\) for each \(k\),
    • \(d_n > 0\), and
    • \(\displaystyle q \,=\, \sum_{k=1}^n \frac{d_k}{(k+1)!}\).
    For example, the factorial digits of \(9/10\) are \(1,2,1,3\), since \[ \frac{9}{10} \;=\; \frac{1}{2!} \,+\, \frac{2}{3!} \,+\, \frac{1}{4!} \,+\, \frac{3}{5!} \] Write a program that takes a rational number between \(0\) and \(1\) as input and outputs its sequence of factorial digits. Demonstrate your program by computing the factorial digits of 29/34.