The term "elementary techniques" in mathematics means use of traditional, very well-understood fields of mathematics like number theory. Other fields of math that require additional assumptions, such as complex analysis, cannot be used in a proof based on elementary techniques. Included in elementary techniques are objects, operations, and relations. Sets, sequences and many types of geometry are not included.
For example, there was a proof of the prime number theorem relying on non-elementary techniques, but in 1949 and 1950 an "elementary proof" by Erdos and Selberg earned Selberg the highest prize in math, the Fields medal.