An elementary proof or elementary technique in mathematics is a proof that uses only real numbers or real analysis rather than the use of complex analysis or reliance on less rigorous axioms, such as the Axiom of Choice. An elementary proof typically cannot be improved by expressing it in simpler form.
The Prime Number Theorem has long been proven using complex analysis (Riemann 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. In contrast, Andrew Wiles' proof of Fermat's Last Theorem is not an elementary proof.
Significance of Elementary Proofs
Elementary proofs are preferred over non-elementary proofs for at least two reasons:
- elementary proofs minimize the underlying assumptions, as in avoiding the assumption that there is a unique, algebraically manipulable square root of negative one
- elementary proofs are typically impossible to simplify further in a logically significant manner