teorema

Teorema is the term used in Italian, Spanish, Portuguese and other Romance languages for a mathematical theorem: a statement that has been proven to be true on the basis of axioms and previously established results. The word derives from Greek theorema, through Latin and later Romance languages. In mathematics, a teorema is a formally proven statement, in contrast to conjectures, which are hypotheses that have not yet been proven.

A teorema is usually presented with a clear statement of hypotheses (assumptions) and a conclusion, followed by a rigorous proof. Proofs may be constructive or non-constructive and employ methods such as direct reasoning, contradiction, contrapositive, mathematical induction, or other techniques. Theorems often rely on auxiliary results called lemmas, and corollaries are results that follow directly and readily from a teorema.

Well-known examples include the Pythagorean theorem, which relates the sides of a right triangle; Fermat’s little theorem, concerning residues in modular arithmetic; and the fundamental theorem of calculus, linking differentiation and integration. These theorems have wide-ranging applications across mathematics and the sciences, providing reliable conclusions derived from core principles.

Beyond mathematics, the term teorema also appears in culture. Teorema is the title of a 1968 film by Pier Paolo Pasolini, illustrating how the concept’s form—proven statements or propositions—has permeated other fields. In everyday use, teorema captures the central idea of truth established through logical proof.