Number theory
While the word "algorithm" goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in "Elements", together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation formula_78,
or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called
"kuṭṭaka" ("pulveriser"), without a proof of correctness.
Algorithmic art
From one point of view, for a work of art to be considered algorithmic art, its creation must include a process based on an algorithm devised by the artist. Here, an algorithm is simply a detailed recipe for the design and possibly execution of an artwork, which may include computer code, functions, expressions, or other input which ultimately determines the form the art will take. This input may be mathematical, computational, or generative in nature. Inasmuch as algorithms tend to be deterministic, meaning that their repeated execution would always result in the production of identical artworks, some external factor is usually introduced. This can either be a random number generator of some sort, or an external body of data (which can range from recorded heartbeats to frames of a movie.) Some artists also work with organically based gestural input which is then modified by an algorithm. By this definition, fractals made by a fractal program are not art, as humans are not involved. However, defined differently, algorithmic art can be seen to include fractal art, as well as other varieties such as those using genetic algorithms. The artist Kerry Mitchell stated in his 1999 "Fractal Art Manifesto":