In the late 1970s it was noted that by evaluating PowerMod[a, n - 1, n]  1 for several random integers a one can with high probability quickly deduce PrimeQ[n] . (In the 1960s it had been noted that one can factor polynomials by filling in random integers for variables and factoring the resulting numbers.)

Such choices are particularly convenient on computers where machine integers are represented by 32 binary digits. … The maximal repetition period of 2 n - 1 can be achieved only if Factor[1 + x + x n , Modulus  2] finds no factors. … In fact, the first known generator for digital computers was John von Neumann 's "middle square method" n  FromDigits[Take[IntegerDigits[n 2 , 10, 20], {5, 15}], 10] In practice this generator has too short a repetition period to be useful.

In 1909 Axel Thue showed that any equation of the form p[x, y]  a , where p[x, y] is a homogeneous irreducible polynomial of degree at least 3 (such as x 3 + x y 2 + y 3 ) can have only a finite number of integer solutions. (He did this by formally factoring p[x, y] into terms x - α i y , then looking at rational approximations to the algebraic numbers α i .) … (For quadratic equations Hasse's Principle implies that if no solutions exist for any n then there are no solutions for ordinary integers—but a cubic like 3x 3 + 4 y 3 + 5 z 3  0 is a counterexample.)

In the late 1940s this procedure was then essentially justified by the idea of renormalization: that since in all possible QED processes only three different infinities can ever appear, these can in effect systematically be factored out from all predictions of the theory.