## 403/503- Homework 1

January 27, 2011

Recall that we defined Nim addition $\oplus$ and Nim multiplication $\otimes$ on the natural numbers ${\mathbb N}=\{0,1,\dots\}$ by setting:

1. $n\oplus m$ is the result of writing $n,m$ in binary and adding without carrying.
2. $\otimes$ is the unique binary operation on ${\mathbb N}$ that is commutative, associative, distributive over $\oplus$ and satisfies:
• $\displaystyle F_n\otimes F_n=\frac32 F_n$ and
• $\displaystyle F_n\otimes m=F_nm$ whenever $m< F_n$. Here, $F_n=2^{2^n}$ for all $n\in{\mathbb N}$, call these numbers “Fermat’s 2-powers”.

The first problem is that it is not quite clear that $\otimes$ is even well-defined (i.e., are there any functions at all that behave as required of $\otimes$? Is there really only one such function?). Begin by checking this, by showing that the rules above give us that $n\otimes m$ is the result of writing $n,m$ as sums of (multiplications by powers of 2) of Fermat 2-powers, and expanding using associativity, distributivity, and the two rules about how to multiply with Fermat 2-powers. (Verify that any $n$ can indeed be written as such a sum in a unique way, and that this indeed shows that $\otimes$ is completely characterized by our description.)

Show that if we set ${\mathbb F}_{2^{2^n}}=\{0,1,\dots,F_n-1\}$, then ${\mathbb N}$ and each ${\mathbb F}_{2^{2^n}}$ are fields (over ${\mathbb Z}_2$, with addition and multiplication given by $\otimes,\oplus$.

In lecture I erroneously mentioned that ${\mathbb N}$ is algebraically closed. This is not the case. For example, show that the equation $x^3+x+1=0$ has no solutions in ${\mathbb N}$ when we interpret “$n$ is a solution” to mean that $(n\otimes n\otimes n)\oplus n\oplus 1=0$.

Nim addition and multiplication were introduced by John Conway. A bit of online searching will give you references for this exercise, but please abstain from looking for them. I will provide references and some additional details once the homework has been turned in.

This set is due February 11 at the beginning of lecture.

## 507- Problem list (IV)

January 21, 2011

For Part III, see here.

(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list. If you have corrections/updates, please email me. Sorry for the delay with posting this.)

## 507- Problem list (III)

January 20, 2011

For Part II, see here.

(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list.)

• The Erdös-Turán conjecture on additive bases of order 2.
• If $R(n)$ is the $n$-th Ramsey number, does $\lim_{n\to\infty}R(n)^{1/n}$ exist?
• Hindman’s problem: Is it the case that for every ﬁnite coloring of the positive integers, there are $x$ and $y$ such that $x$, $y$, $x + y$, and $xy$ are all of the same color?
• Does the polynomial Hirsch conjecture hold?
• Does $P=NP$? (See also this post (in Spanish) by Javier Moreno.)
• Mahler’s conjecture on convex bodies.
• Nathanson’s conjecture: Is it true that ${}|A+A|\le|A-A|$ for “almost all” finite sets of integers $A$?
• The (bounded) Burnside’s problem: For which $m,n$ is the free group $B(m,n)$ finite?
• Is the frequency of 1s in the Kolakoski sequence asymptotically equal to $1/2$? (And related problems.)
• A question on Narayana numbers: Find a combinatorial interpretation of identity 6.C7(d) in Stanley’s “Catalan addendum” to Enumerative combinatorics.

January 20, 2011

Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 11:40 am – 12:30 pm.
Place: Mathematics/Geosciences building, Room 124.
Office Hours: MF 10:40-11:30 am.
Text: Axler, Sheldon. Linear algebra done right. Springer, 2nd edition (1997).

Contents: Math 403/503 is intended to be a second course in linear algebra, where an abstract approach emphasizing the role of linear transformations is preferred to a more computational approach based on properties of matrices. From Course Description in the Department’s site:

Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form.

Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.