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

.