Theorem

: A statement that follows almost immediately from a proven theorem with little or no additional proof required. Famous Examples of Theorems

Theorems form the backbone of fields ranging from basic geometry to advanced computer science and cryptography. Core Concept In a right triangle, the square of the hypotenuse ( ) equals the sum of the squares of the legs ( Fundamental Theorem of Calculus theorem

: The "given" or foundational statements that are accepted as true without proof. All proofs eventually trace back to these. : A statement that follows almost immediately from

The distinction between a conjecture and a theorem is the existence of a proof. For example, the —which states that every even integer greater than 2 is the sum of two primes—has been tested for trillions of numbers and appears true, but because it lacks a formal proof, it remains a conjecture rather than a theorem. The Evolution of Proof All proofs eventually trace back to these

: The logical argument that demonstrates why a theorem must be true. Modern proofs must follow strict rules of inference to be accepted by the mathematical community.

: A "helper" result. Lemmas are smaller theorems used as stepping stones to prove a larger, more significant result.