You are correct in the .put on the map. I forgot to add that when typing it.

Can you add a few lines of debug code and turn on your debug logs. Then post the values of the debug code here to see, or PM them to me?

Update the code to this to do some debuging:

//id will be project_c ID so we can look up ProjectAnalysis by Project__c ID
Map<id,ProjectAnalysis__c> mapOfPA = New Map<id, ProjectAnalysis>();

//Build a map of Project__c to ProjectAnalysis using Project__c ID
for(ProjectAnalysis oPA : listPA)

      mapOfPA.put(oPA.Project__c, oPA);


system.debug('################### mapOfPA = ' + mapOfPA);


      //This loops through the new tirgger
for(Project__c nP : trigger.new){   

system.debug('################## nP ID = ' nP.ID);

   //This gets the record by ID before it was changed
   Project__c oProject = trigger.oldMap.get(nP.Id);  


system.debug('#################### oProject APM = ' + oProject.Amount_Per_Month__c + ' nP.Amount_Per_Month__c);
      
         if (oProject.Amount_per_month__c!= nP.Amount_per_month__c){          

               mapOfPA.get(np.id).Amount_Per_Month__c = nP.Amount_Per_Month;


system.debug('################ mapOfPA APM = ' + mapOfPA.get(nP.id).Amount_Per_Month);

               hasChanged = true;

          }
      }
    }


if(hasChanged)

  update mapOfPA.values()
}   

The ############### are just so you can find it in the log eaisly. Just in case you did not know, you will have to go to Setup -> -> Data Monitoring - > Debug Logs and then click NEW and add your Name. Then when the trigger runs, it will be output to the debug log.

Sorry for all the hoops, just trying to figure this out with ya. The debug out put will be helpful in understanding what it is actually doing.