Writing a in terms of distinct factors as r s , {r + s, r - s}/2 gives a solution if it yields integers—which happens when Abs[a] > 4 and Mod[a, 4] ≠ 2 . • x 2  a y 2 + 1 (Pell equation).

A major mathematical achievement in the 1980s was the complete classification of all possible so-called simple finite groups that in effect have no factors.

Several other definitions used in specific fields in the 1960s and 1970s were also based on sizes of descriptions: examples were optimal orders of models in systems theory, lengths of logic expressions for circuit and program design, and numbers of factors in Krohn–Rhodes decompositions of semigroups.

In quantum field theory the whole concept of measurement is much less developed than in quantum mechanics—not least because in field theory it is much more difficult to factor out subsystems, and so to avoid having to give explicit descriptions of measuring devices.

(An idealized soap film or other minimal surface extremizes the integral of the intrinsic volume element Sqrt[Det[g]] , without a RicciScalar factor.)

The way the path integral for a quantum field theory works, each possible configuration of the field is in effect taken to make a contribution Exp[  s/ ℏ ] , where s is the so-called action for the field configuration (given by the integral of the Lagrangian density—essentially a modified energy density), and ℏ is a basic scale factor for quantum effects (Planck's constant divided by 2 π ).

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.

(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 .)