Math 328K (Daniel Allcock)
Homework 2, due Friday Feb. 5, 2016
As usual, a problem which is just a statement means that you must prove it.
Problem 1. The equation x3 − x = y 2 + 1 has no integer solutions. (Hint: find a
modulus m such that there are no solutions modulo m. Once you find such an m,
this is like problems 5–6 on homework 1. To find m, just try some small numbers.
I promise you won’t have to try too many of them.)
Problem 2. Let m ∈ Z. Show that congruence modulo m is an equivalence relation.
In other words, taking “equivalence” to mean “congruence modulo m”, you must
show the following.
(i) each integer a is equivalent to itself;
(ii) for any two integers a and b, a is equivalent to b if and only if b is equivalent
to a; and
(iii) for any three integers a, b and c, if a is equivalent to b and b is equivalent
to c, then a is equivalent to c.
Now that we know congruence modulo m is an equivalence relation, we can make
the definition: if a ∈ Z, then the phrase “a mod m” means the equivalence class
containing a (that is, the set of all integers equivalent to a). Caution: technically,
this is different from what you might have used “mod” for in the past, especially
in computer programming. There something like “45 mod 7” is a number (in this
case 3), whereas for us, “45 mod 7” means a set of numbers.
Problem 3. (a) Write down the set which is the literal meaning of “3 modulo 11”.
(b) Observe that it is exactly the same set as “−19 modulo 11”. So
“3 mod 11”
=
“−19 mod 11”
is a genuine equality. (Sometimes we say informally that 3 and −19 are equal
mod 11.)
(c) Write down the set which is the literal meaning of “−4 mod −7”. (This example is meant for the computer programmers: most compilers will accept (-4)%(-7),
but the meaning is implementation dependent, which means you should never use
it in a program. But the mathematical definition is unambiguous. If you don’t
know what I’m talking about then just ignore this remark.)
The set of all equivalence classes of integers mod m is written Zm , pronounced “Z
mod m”. If a ∈ Z then we also introduced the notation a for a’s equivalence class.
(This notation is not in the book.) Remember that this notation only has meaning
after the modulus has been fixed, so be sure to specify the modulus whenever you
use it. Setting up a number system has three steps: saying what it is as a set, then
defining the basic operations, and then establishing the usual properties of these
operations. We did step 1 in class and repeated it above, and also did Step 2 in
class; the next problem is step 3.
Repeating from class the definitions of the operations: if a, b ∈ Z then a + b is
defined as a + b and a · b is defined as ab, and checked that these operations are
well-defined. (Note that a + n and ab use + and · in Z (not Zm ). Since we are
taking these as already defined, the definition is not circular.)
Problem 4. Addition in Zm is commutative and associative, multiplication is also,
and the distributive law holds: a(b + c) = ab + a c for all a, b, c ∈ Zm . (Hint: Even
though there are 5 things to prove, they are all quick once you figure out what is
needed. Don’t be intimidated. The important thing is to phrase the proof carefully,
so properties of the integers imply the facts you want.)
Problem 5. At first it seems reasonable to mimic the definition of even integers and
call a number in Zm “even” if it has the form 2 · x for some x ∈ Zm .
(a) Under this definition, all members of Z9 are even!
(b) If m is even and a ∈ Z, then a is even as an element of Zm if and only if a
is even as an integer.
(Remark: the only thing special about 9 in (a) is that it is odd. The strangeness
there feels so wrong that we will avoid calling elements of Zm “even” or “odd” if
m is odd. When m is even, part (b) says that the word “even” behaves reasonably,
and we call a member of Zm “odd” if it is not even.)
The next problem continues developing the idea that there are not just a few
fixed number systems in the world. There are lots, and they aren’t just given to
you—you can build new ones from old ones whenever it seems useful. You will see
the Gaussian integers again on the next homework.
A Gaussian integer means an ordered pair (a, b) of integers. Addition is defined
by setting (a, b) + (c, d) equal to (a + c, b + d), where (c, d) is another such pair.
Similarly, multiplication is defined by setting (a, b) · (c, d) equal to (ac − bd, ad +
bc). Because the definition doesn’t involve any choices, like it did when defining a
number system in terms of an equivalence relation, we don’t have to check welldefinedness. There is more about the Gaussian integers late in the book. But for
the problems I’ll give you, you don’t need anything more than the definitions and
your brain.
Problem 6. You can assume without proof that addition in the Gaussian integers
is commutative and associative. Prove that multiplication is also, and that the
distributive law holds.