I am relatively new to writing proofs. I'm working through Gallian's
Contemporary Abstract Algebra. As a new proof writer, I occasionally feel a bit awkward . . . I get the sense I am "running in place," that is, obscuring the proof with too many steps, details, and general hem-hawing.
Below I have given a proposition and my proof. I feel comfortable with the
logic of the proof, but I look at it and think, "This could
really be written better."
However, that's where I'm drawing a blank. So I called in the experts.
Can this proof be made "pretty"?
Proposition: For all integers
,
and
are relatively prime.
Proof. If
,
, and
and
are relatively prime, then
. Since
for some integers
and
, this relation can be written in terms of
as
. Therefore
. This corresponds to a relation of the form
. Because 3 and 4 are relatively prime, we can choose
and
so that
, and therefore, the
term must be zero for those same choices. If
and
, then
. Thus,
and
will be relatively prime for any choice of
.