Mechanisms that generally seem able to give α ≃ 1 include random walks with exponential waiting times, power-law distributions of step sizes (Lévy flights), or white noise variations of parameters, as well as random processes with exponentially distributed relaxation times (as from Boltzmann factors for uniformly distributed barrier heights), fractional integration of white noise, intermittency at transitions to chaos, and random substitution systems.

Then starting in the early 1600s various methods for factoring were developed, and conjectures about formulas for primes were made.

Spin values can be thought of as specifying which irreducible representation of the group of symmetries of spacetime is needed to describe a particle after momentum has been factored out.

The maximal repetition period of 2 n - 1 can be achieved only if Factor[1 + x + x n , Modulus  2] finds no factors. … For a register of size n the maximal period of 2 n -1 is obtained whenever x n + Apply[Plus, x taps - 1 ] is one of the EulerPhi[2 n - 1]/n primitive polynomials that appear in Factor[Cyclotomic[2 n - 1, x], Modulus  2] . … The simplest example uses the rule n  Mod[n 2 , m] It was shown that for m = p q with p and q prime the sequence Mod[n, 2] was in a sense as difficult to predict as the number m is to factor (see page 1090 ).

(In the 1960s it had been noted that one can factor polynomials by filling in random integers for variables and factoring the resulting numbers.)

Sometimes nonzero size is taken into account by inserting additional interaction parameters—as done in the 1950s with magnetic moments and form factors of protons and neutrons. … But the development of renormalization in the 1940s showed that these infinities could in effect just be factored out.

Already in 1882 George FitzGerald and Hendrik Lorentz noted that if there was a contraction in length by a factor Sqrt[1 - v 2 /c 2 ] in any object moving at speed v (with c being the speed of light) then this would explain the result.

It is not known whether problems such as integer factoring or equivalence of networks under relabelling of nodes (graph isomorphism) are NP-complete.

And even in this case the set of possible solutions for x and y in the pictures below looks fairly complicated—though after removing common factors, they are in fact just given by {x  r 2 - s 2 , y  2 r s, z  r 2 + s 2 } .

Unlike for continuous mathematical functions, known algorithms for number theoretical functions such as FactorInteger[x] or MoebiusMu[x] typically seem to require a number of operations that grows faster with the number of digits n in x than any power of n (see page 1090 ).