It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." 


I have two closely related questions.


1. Is my understanding of the usage of "elementary" correct?
2. It appears that advanced techniques from other areas (e.g. algebra) are allowed, just not complex variables. Are there historical reasons for why complex analysis singled out as a tool to avoid?


NB: I'm asking about how "elementary" usually **is** defined and why, not how it **should be** defined.



Definition of elementary number theory
Your usage of "elementary" is correct; your definition is the one that most number theorists would use. You don't have to take my word for it however; just consider the first sentence of [Selberg's Elementary Proof of the Prime Number Theorem](http://www.jstor.org/pss/1969455):


*In this paper will be given a new proof of the prime-number theorem, which is elementary in the sense that it uses practically no analysis, except the simplest properties of the logarithm.*


Ironically, of the many known proofs of the prime-number theorem, this *elementary* proof ranks as one of the most complicated.


