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Rational numbers require only division (or solving linear equations), while algebraic numbers require solving polynomial equations. … One can also ask what numbers can be generated by integrals (or by solving differential equations). … One can also consider numbers obtained from infinite sums (or by solving recurrence equations).

LFSR cryptanalysis
Given a sequence obtained from a length n LFSR (see page 975 )
Nest[Mod[Append[#, Take[#, -n] . vec], 2] &, list, t]
the vector of taps vec can be deduced from
LinearSolve[Table[Take[seq, {i, i + n - 1}], {i, n}], Take[seq, {n + 1, 2n}], Modulus 2]
(An iterative algorithm in n taking about n 2 rather than n 3 steps was given by Elwyn Berlekamp and James Massey in 1968.)

Linear and nonlinear systems
A vast number of different applications of traditional mathematics are ultimately based on linear equations of the form u m . v where u and v are vectors (lists) and m is a matrix (list of lists), all containing ordinary continuous numbers. … However, it so happens that even in this case v can still be found fairly straightforwardly using LinearSolve[m, u] .

Recurrence relations
The rules for the sequences given here all have the form of linear recurrence relations. An explicit formula for the n th term in each sequence can be found by solving the algebraic equation obtained by applying the replacement f[m_] t m to the recurrence relation. … Standard examples of recursive sequences that do not come from linear recurrence relations include factorial
f[1] = 1; f[n_] := n f[n - 1]
and Ackermann functions (see below ).

.)
• Is there a solution of a certain size to an integer linear programming problem?
… (See page 984 .)
• Is there a string of some limited length that solves a correspondence problem?
… (The latter is like a network with only 2 connections at each node.)
• Solving quadratic Diophantine equations a x 2 + b y c is NP-complete, but solving linear ones a x + b y c is in P.
• Finding a minimum energy configuration for a 2D Ising spin glass in a magnetic field is NP-complete, but is in P if there is no magnetic field.
• Finding the permanent of a matrix is NP-complete, but finding its determinant is in P.

Mathematical equations that have the property of linearity are usually fairly easy to solve, and so have been used extensively in pure and applied science.

The objects that appear are solutions to linear recurrences of order s , and in general involve s th roots. … Solve[z 2 h z + 1] and h = 1/x + 1 + x .

What the values for these blocks should be can be found by solving a system of linear equations; that a solution must exist can be seen by looking at the de Bruijn network (see page 941 ), with nodes labelled by size b + 2r - 1 blocks, and connections by value differences between size b blocks at the center of the possible size b + 2r blocks.

The most common are ones based on repetition or iteration, classic examples being Euclid's algorithm for GCD (page 915 ), Newton's method for FindRoot and the Gaussian elimination method for LinearSolve .

Among such computations are Plus , Times , Divide , Det and LinearSolve for integers, as well as determining outcomes in additive cellular automata (see page 609 ).