Mathematician by Result

A fundamental relation between the lengths of the sides of a right-angled triangle.

Any consistent formal system in which a certain amount of arithmetic can be carried out is incomplete. Moreover, the consistency of this system cannot be proved within the system.

The area of a circle is equal to pi multiplied by the square of the radius.

There exist infinitely many prime numbers.

No positive integers a, b, c satisfy the equation a^n + b^n = c^n for any integer n greater than 2.

An abstract model of computation, where symbols on an infinite tape are manipulated according to a set of rules.

An identity relating five fundamental mathematical constants: 0, 1, e, pi and i. Famously described by physicist Richard Feynmann as "the most remarkable formula in mathematics."

The assertion that there exists no infinite set with cardinality strictly between that of the natural numbers and that of the real numbers. Equivalently, 2^{ℵ_0}=ℵ_1.

Let a and b be integers with greatest common divisor d. Then, there exist integers x and y such that ax + by = d.

For a subgroup H of any finite group G, the order of H divides the order of G.

A formula generalising the expansion of (x+y)^n for any positive integer n. The coefficient of each term in the expansion is known as the binomial coefficient.