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YouTube Videos – Algorithm and Related Articles

Alan Turing's statue at Bletchley Park

The example-diagram of Euclid's algorithm from T.L. Heath (1908), with more detail added. Euclid does not go beyond a third measuring, and gives no numerical examples. Nicomachus gives the example of 49 and 21: "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7." Heath comments that, "The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'."(Heath 1908:300).

"Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2–4. Depending on the two numbers "Inelegant" may compute the g.c.d. in fewer steps than "Elegant".

Image: Euclid's algorithm structured blocks 1