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 .