Elementary proof

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The term "elementary proof" or "elementary techniques" in mathematics means proofs that use only real numbers or real analysis rather than the use of complex analysis.[1] Elementary proofs cannot be broken down into smaller proofs of the same proposition.


The prime number theorem has long been proven using complex analysis (Riemann's zeta function), but in 1949 and 1950 an elementary proof by Paul Erdos and Atle Selberg earned Selberg the highest prize in math, the Fields Medal.


Sources:
  1. http://mathworld.wolfram.com/ElementaryProof.html