# Math poetry: prime numbers and Nemo

“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 $6 \times 10^{23}$ of chemistry, call it $10^{24}$ — 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 $10^{24}$, this mole. The relevance is that a task such as trial dividing an integer $N \simeq 10^{50}$ 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 $10^{16}$ to $10^{18}$ 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 $10^{18}$ 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 $10^{18}$ 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]