FINAL EXAMINATION: MATH 406
SPRING 2013
This examination, worth a total of 110 points, has five questions. All format and
content requirements applying to homework assignments also apply to this test.
Please start the presentation of your work on each question on a new page, with
a full statement of the question. You may discuss the problems on this test with
fellow enrollees in this course only, and you may consult any published literature on
the subject. However, you must write up your solutions in your own words, and you
must properly attribute ideas and information gained from other sources, including
discussions with other enrollees in the course. For the purposes of this test it is
more important to produce a scholarly document that demonstrates your grasp of
the material and your ability to properly and thoroughly explain the solutions to
these problems, than it is to be the inventor of these solutions. Answers without
accompanying exposition that shows that you can coherently and logically explain
the mathematical reasons justifying your answers will not receive any credit. The
examination is due by 11:30 am on May 16, 2013.
Notation and conventions:
(I) For functions f and g with domain the set of positive integers, f ]g denotes the
function with value at n given by
X
n
f (d) · g( ).
d
d|n
(II) In the questions below, p denotes a prime number larger than 3.
Question 1 (20 points) Let n be a positive integer, and let D be a positive integer
that is not a perfect square. Prove that if
x2 − D ∗ y 2 = n
has a solution (x, y) ∈ Z × Z, then it has infinitely many such solutions.
Question 2 (20 points) Put P3 = {n ∈ Z : (∃(x, y) ∈ Z × Z)(x2 − 3 ∗ y 2 = n)}
Prove that P3 is closed under multiplication.
Question 3 (35 points) For an integer n consider the Pell-like equation
x2 − 3 ∗ y 2 = n where variables x and y range over the integers. Prove that for
prime numbers n the following are equivalent (10 points for (a) ⇒ (b), 10 points
for (b) ⇒ (c) and 15 points for (c) ⇒ (a)):
(a) x2 − 3 ∗ y 2 = n has a solution.
(b) x2 − 3 ∗ y 2 = n has infinitely many solutions.
(c) n mod 12 = 1.
1
2
SPRING 2013
Question 4 (20 points) Let µ be the Möbius function and let φ be Euler’s totient
function.
(a) Find the ]-inverse of µ. (10 points)
(b) Find the ]-inverse of φ. (10 points)
Question 5 (15 points) Let σ be the function that assigns to a number n the sum
of all divisors of n, minus n. Consider the following process:
For a positive integer x0 = x compute x1 = σ(x0 ), x2 = σ(x1 ),
· · · , xk+1 = σ(xk ), · · ·
(a) Is it true that there is for each positive integer n a k such that with
x0 = n, for all m > k we have xm = xk ? (5 points)
(b) Is it true that there is for each positive integer n a k such that with
x0 = n, for all m we have xm ≤ k? (10 points)