403/503- Homework 1

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.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: