I was grading some homework from a Survey of Mathematics course. They were asked to verify that Goldbach's conjecture holds for the first 15 even numbers greater than or equal to 4. A couple of students wrote "solutions" such as 12 = 11 + 1. It struck me in passing as an interesting error. What if the definition of primes was modified so that 1 would count as a prime? Then Goldbach's conjecture could be stated simply as every even natural number is the sum of two primes (since of course 2 = 1 + 1). More formally (and without modifying the definition of a prime): Every even number greater than 4 is either the sum of two primes or is the successor of a prime. Surely this weakening of Goldbach's conjecture has been studied, but I wasn't able to find any discussion of it.
Has this variation of Goldbach's conjecture been studied? If so, what is known about it?