Problem 1
c.f. Problem 7.1
Calculate \(d(42)\), \(\sigma(42)\), \(d(420)\), and \(\sigma(420)\).
Assignment 7
Due Thursday, October 27, 2022
Problem 2
c.f. Problem 7.4
Calculate \(d(10116)\), \(\sigma(10116)\), \(d(100116)\), and \(\sigma(100116)\).
Problem 3
c.f. Problem 7.5
Show that \(\sigma(n)\) is odd if \(n\) is a power of \(2\).
Problem 4
c.f. Problem 7.6
Show that if \(f(n)\) is multiplicative, then so is \(f(n)/n\).
Problem 5
c.f. Problem 7.7
What is the smallest integer \(n\) such that \(d(n)=8\)? Such that \(d(n)=10\)?
Problem 6
c.f. Problem 7.9
In \(1644\), Mersenne asked for a number with \(60\) divisors. Find one smaller than \(10000\).
Problem 7
c.f. Problem 7.13
If \(n\) is a square, show that \(d(n)\) is odd.
Problem 8
c.f. Problem 7.17
If \(n\) is odd, how many integer solutions does \(x^2-y^2=n\) have?
Solution
The equation \(x^2-y^2=n\) is equivalent to \((x+y)(x-y)=n\). For every integer divisor \(a\) of \(n\), we want to solve the two simultaneous equations \[\begin{align*} x+y &= a\\ x-y &= b. \end{align*}\]
By linear algebra, there is a unique solution over the rationals: \(x=\frac{1}{2}(a+b)\) and \(y=\frac{1}{2}(a-b)\). However, we need to verify that \(x\) and \(y\) are both integers. The product of even number with any other integer is still even. Since \(n\) is odd, if \(n=ab\) for integers \(a,b\), then both \(a\) and \(b\) must be odd. The sum or difference of two odd numbers is even, so \(x=\frac{1}{2}(a+b)\) and \(y=\frac{1}{2}(a-b)\) are always integers.
Thus, there is exactly one solution to \(x^2-y^2=n\) for every divisor of \(n\). There is exactly one positive divisor for every negative divisor. Since there are \(d(n)\) positive divisors, the number of integer solutions is \(2d(n)\).