Engineering Mathematics

N+1 Tests and the Lucas-Lehmer Test

Primality tests using the factorisation of one more than the candidate, and the Lucas-Lehmer test for Mersenne numbers.

New Linear Transformations From Old

Sums, scalar multiples and compositions of linear transformations are linear. The set of all such maps is a vector space, and matrix operations mirror each construction.

New Matrix Representations From Old

Sums, scalar multiples and compositions of linear transformations become sums, multiples and products of their matrix representations: multiplication is composition.

Newton Polygon Methods for Prime Decomposition

Using Newton polygons to decompose primes locally, handling the cases where factoring modulo p is insufficient.

Nonsingular Matrices and the Identity Matrix

Definition of a nonsingular square matrix via the trivial null space, the identity matrix, the row-reduction test and unique solutions for any right-hand side.

Engineering Mathematics13 Aug 2026 · 21 min read

Nonsingular Matrices Are Invertible

Proof that a square matrix is nonsingular exactly when it is invertible, via products with singular matrices and the sufficiency of a one-sided inverse.

Engineering Mathematics13 Aug 2026 · 15 min read

Nonsingular Matrix Equivalences — Applications and Numerical Practice

Thirteen equivalent conditions on a square matrix: nonsingular, row-reduces to the identity, trivial null space, invertible, full rank, nonzero determinant.

Engineering Mathematics13 Aug 2026 · 18 min read

Nonsingular Matrix Equivalences — Foundations and Proof Network

Thirteen equivalent conditions on a square matrix: nonsingular, row-reduces to the identity, trivial null space, invertible, full rank, nonzero determinant.

Null Space of a Matrix

The null space N(A) is the set of all vectors annihilated by a matrix: its definition, how to compute a spanning set from free variables, and why its dimension is n minus the rank.

Null Spaces, Spans and Linear Independence

Row reduction yields n-r linearly independent vectors spanning the null space of a matrix: a minimal, non-redundant basis for every homogeneous solution.

Number Field Sieve: Polynomial Selection and Structure

How the number field sieve achieves its complexity, why polynomial selection matters so much, and the role of number field arithmetic.

Number Fields: Definition and Basic Properties

Number fields as finite extensions of the rationals, their embeddings and signature, and the presentation on which all computation depends.

Number Theory Software Packages

The computer algebra systems and libraries implementing these algorithms, what each is suited to, and why implementing from scratch is usually the wrong choice.

Operations on Subspaces and Modules

Sum, intersection and quotient of subspaces, and how the analogous operations on modules over the integers require normal forms.

Orders in Number Fields

Orders as subrings that are full-rank lattices, the equation order, the maximal order, and the index that separates them.

Orthogonal and Unitary Matrices

Matrices whose adjoint is their inverse: the condition A*A = I, orthonormal columns, preservation of inner products and norms, adjoints and Hermitian matrices.

Orthogonal Vectors and Orthonormal Sets

Orthogonality of complex vectors, orthogonal and orthonormal sets, why an orthogonal set of non-zero vectors is linearly independent, and how to test one.

Orthonormal Bases and Coordinates

How an orthonormal basis turns coordinate extraction into a single inner product, why every subspace has one, and how orthogonal matrices supply ready-made orthonormal bases.

Orthonormal Diagonalization

Hermitian and real symmetric matrices diagonalise with a unitary or orthogonal change of basis: real eigenvalues, orthogonal eigenvectors and the spectral decomposition.

p-adic Root Finding and Newton Polygons

Finding roots in p-adic fields by lifting, and reading ramification structure off the Newton polygon of a polynomial.

Pairwise Independence and Universal Hash Families

Constructing pairwise independent hash families over finite fields and why the weaker independence suffices.

Parameter Sizes, Records and Live References

Where to obtain current key size recommendations, factoring records and post-quantum guidance, and why they are not reproduced here.

Engineering Mathematics13 Aug 2026 · 20 min read

Particular Solutions and Homogeneous Solutions

Every solution of a consistent linear system is one particular solution plus an element of the null space, so solution sets are affine translates of a subspace.

Perfect Power Testing and Prime Power Factoring

Detecting whether an integer is a perfect power, extracting the root, and why this precedes general factoring.

Polynomial Congruences

Congruences of polynomials modulo a fixed polynomial, and the parallel with integer congruences.

Polynomial Division with Remainder

Euclidean division of polynomials, pseudo-division over a ring without inverses, and the coefficient growth pseudo-division introduces.

Polynomial Factorisation: Overall Strategy

The three-stage pipeline used to factor polynomials over finite fields and the integers, and why the stages are ordered as they are.

Polynomial GCD over a Unique Factorisation Domain

Computing polynomial GCDs over the integers, the growth problem in remainder sequences, and the modular approach that sidesteps it.

Polynomial Modular Inverses

Inverting a polynomial modulo another using the extended Euclidean algorithm, and the finite field application.

Polynomial Multiplication Strategies

Schoolbook, Karatsuba and evaluation-interpolation methods for polynomial multiplication, and where the crossovers lie.

Polynomial Quotient Algebras

The structure of F[X]/(f), its basis, arithmetic, and the decomposition when f is reducible.

Polynomial Representation and Storage

Dense and sparse polynomial representations, coefficient domains, and the normalisation invariants every implementation must maintain.

Polynomials and Matrices

Evaluating a polynomial at a square matrix: matrix powers, why the constant term becomes a multiple of the identity, and why the factored form gives the same answer.

Polynomials versus Polynomial Functions

The distinction between a formal polynomial and the function it induces, and why the two differ over finite rings.

Possibilities for Solution Sets of Linear Systems

A linear system has no solution, exactly one, or infinitely many — never two. The trichotomy, its geometric reading, its proof, and what each case means in practice.

Pre-Images of a Linear Transformation

The pre-image of a vector under a linear transformation: when it is empty, why a non-empty pre-image is a translate of the kernel, and how it partitions the domain.

Primality Certificates and Independent Verification

What a primality certificate is, why verification is cheaper than production, and what a certificate does and does not guarantee.

Primality Versus Factoring: Framing the Problems

Why proving compositeness is easy, proving primality is harder, and factoring is harder still — and what this asymmetry means in practice.

Prime Decomposition in Quadratic Fields

How rational primes split, remain inert or ramify in a quadratic field, decided entirely by the Kronecker symbol.

Prime Decomposition when p Does Not Divide the Index

Decomposing a prime that does not divide the index, by factoring the defining polynomial modulo that prime.

Prime Decomposition: Theory and Ramification

How rational primes factor in the maximal order, ramification indices and residue degrees, and the degree relation that constrains them.

Primes in Arithmetic Progressions

Dirichlet's theorem on primes in arithmetic progressions, equidistribution across residue classes, and computational consequences.

Probabilistic Algorithms: Foundations

The model of randomised computation, Las Vegas and Monte Carlo algorithms, and what a probabilistic guarantee means.

Engineering Mathematics13 Aug 2026 · 26 min read

Proof Techniques in Linear Algebra

The standard proof patterns of linear algebra: definitions as two-way implications, set equality, equivalences, contrapositives, uniqueness, chains of identities and TFAE cycles.

Properties of Determinants

Transpose invariance, multiplicativity over matrix products, and the equivalence between a zero determinant and singularity, with the consequences for inverses and rank.

Properties of Eigenvalues and Eigenvectors

Eigenvalue theorems: independence for distinct eigenvalues, zero eigenvalues and singularity, and the spectra of powers, polynomials, inverses and transposes.

Engineering Mathematics13 Aug 2026 · 21 min read

Properties of Matrix Inverses

Algebraic rules for matrix inverses: uniqueness, the reversal rule for products, inverse of an inverse, inverse of a transpose and of a scalar multiple.

Properties of Matrix Multiplication

Associativity, distributivity, the identity and zero matrices, scalar rules, and how transposition, conjugation and the inner product interact with matrix products.