“How many computing operations have been performed by all machines across all of world history?” … To put it roughly, right around the turn of the century (2000 AD), about one mole — that is, the Avogadro number of chemistry, call it — is the total operation count for all machines for all of history. In spite of the usual mystery and awe that surrounds the notion of industrial and government super-computing, it is the huge collection of personal computers that allows this , this mole. The relevance is that a task such as trial dividing an integer directly for prime factors is hopeless in the sense that one would essentially have to replicate the machine effort of all time. To convey an idea of scale, a typical instance of the deepest factoring or primality-proving runs of the modern era involves perhaps to machine operations. Similarly, a full-length graphically rendered synthetic movie of today—for example, the 2003 Pixar/Disney movie Finding Nemo—involves operation counts in the range. It is amusing that for this kind of Herculean machine effort one may either obtain a single answer (a factor, maybe even a single “prime/composite” decision bit) or create a full-length animated feature whose character is as culturally separate from a one-bit answer as can be. It is interesting that a computational task of say operations is about one ten-millionth of the overall historical computing effort by all Earth-bound machinery.

[excerpt from Crandall & Pomerance, “Prime Numbers – A Computational Perspective”, 2nd edition, Springer (2005), p. 5]

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