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.
Armatoloi
The word "armatolos" first appeared in the 15th century during Venetian times. It is derived from a medieval loan from Latin "arma" ('weapon'), probably via Greek αρματολόγος ('someone who deals with arms', 'an armed person') → αρματολόος → αρματολός. According to an older hypothesis, the development of the word may also have been influenced by a conflation with the similar-sounding αμαρτωλός ("sinner"; cf. "hamartia"), which may have been associated with the topic of armed bands through phrases such as "αμαρτωλοί/αρματολοί και κλέφτες" (meaning "sinners and thieves", but also "armatoloi and klephts"). Owing to the parallelism with "αμαρτωλός", the word was also sometimes spelled as "αρματωλός", with the letter omega.