Can two positive ions be next to each other in a formula?
Can two positive ions be next to each other in a formula?


Yes, there are many examples of this, $\ce{HNO3}$ for instance


  So Arsenic must have a charge of 3+ for all the charges to add up to
  be 0? Correct?


Yes, that's correct


  However, my professor keeps telling me to keep the positive oxidation
  numbers away from each other. Is that a rule when writing a formula?


It's not a rule that I'm familiar with.  Here is a link to a nice, concise listing of the rules for writing formulas for inorganic compounds.


  I thought it only applied when drawing Lewis structures or resonance
  structures.


Yes, it certainly applies in those two cases